Article of impeachment

If there is evidence that the president is hindering the investigation into his own conduct then the House of Representatives could write and article of impeachment on that. Another answer by PoloHoleSet goes into more detail quoting the chair of the House Intelligence Committee stating that discouraging witnesses to appear and the withholding of documents will be considered as evidence of obstruction.

A similar article was drafted for President Nixon so there is precedent (emphasis mine):

In his conduct of the office of President of the United States, Richard M. Nixon, contrary to his oath faithfully to execute the office of President of the United States and, to the best of his ability, preserve, protect, and defend the Constitution of the United States, and in violation of his constitutional duty to take care that the laws be faithfully executed, has failed without lawful cause or excuse to produce papers and things as directed by duly authorized subpoenas issued by the Committee on the Judiciary of the House of Representatives on April 11, 1974, May 15, 1974, May 30, 1974, and June 24, 1974, and willfully disobeyed such subpoenas. The subpoenaed papers and things were deemed necessary by the Committee in order to resolve by direct evidence fundamental, factual questions relating to Presidential direction, knowledge or approval of actions demonstrated by other evidence to be substantial grounds for impeachment of the President. In refusing to produce these papers and things Richard M. Nixon, substituting his judgment as to what materials were necessary for the inquiry, interposed the powers of the Presidency against the the lawful subpoenas of the House of Representatives, thereby assuming to himself functions and judgments necessary to the exercise of the sole power of impeachment vested by the Constitution in the House of Representatives.

In all of this, Richard M. Nixon has acted in a manner contrary to his trust as President and subversive of constitutional government, to the great prejudice of the cause of law and justice, and to the manifest injury of the people of the United States.

Wherefore, Richard M. Nixon, by such conduct, warrants impeachment and trial, and removal from office.

Inherent Contempt

A different measure, inherent contempt, is explained by the Washington Post (quote below, emphasis mine). Such a measure would have an effect on Trump's presidency when we are talking about high level officials in his administration. While the president himself cannot be simply arrested, his officials may and they may impede his ability to govern.

We know of one tool Congress still has, and it’s pretty blunt: inherent contempt. Some top Democrats have been talking about using this for weeks. It is a long-dormant power for Congress to fine or jail officials who don’t comply. It hasn’t been used in over a century, but when it has, Congress has detained administration officials for not complying with it. Congress doesn’t have its own jail, so lawmakers could try to use the D.C. jail. As you can see, this idea gets pretty extreme pretty quickly.

The idea of inherent contempt isn’t just on the table for the most liberal Democrats. A group of seven freshman Democrats with military records and/or national security backgrounds wrote an op-ed in The Washington Post last week saying Trump’s actions, if true, are impeachable, and they brought up this idea.

“We call on our colleagues in Congress to consider the use of all congressional authorities available to us, including the power of “inherent contempt” and impeachment hearings, to address these new allegations, find the truth and protect our national security,” they wrote.

_{Attribution: https://www.washingtonpost.com/politics/2019/10/01/what-can-congress-do-if-mike-pompeo-wont-cooperate-with-its-impeachment-inquiry/}

Inherent contempt may also be impossible for a president to solve by pardoning (which technically he can do with criminal charges). From the Washington Post's Congress's Power To Compel (emphasis mine):

Yet under historic and undisturbed law, Congress can enforce its own orders against recalcitrant witnesses without involving the executive branch and without leaving open the possibility of presidential pardon.

And a Supreme Court majority would find it hard to object in the face of two entrenched legal principles.

First is the inherent power of Congress to require testimony on matters within its legislative oversight jurisdiction.

So long as Congress is investigating issues over which it has the power to legislate, it can compel witnesses to appear and respond to questions. That power has been affirmed over and over in prosecutions for contempt.

In modern times, this congressional power has been enforced by referring contempt cases to the U.S. attorney for the District of Columbia for indictment and prosecution. That, of course, is the rub. It allows the president to exercise his plenary power under the Constitution to issue pardons "for offenses against the United States."

But no law says that indictment and prosecution by the Justice Department is the exclusive means to enforce congressional prerogative.

Indeed, in an 1895 case ( United States v. Chapman), the defendant unsuccessfully argued that Congress could not have such cases of contempt prosecuted through the courts but must punish such defiance on its own, without judicial assistance. The U.S. Court of Appeals for the District of Columbia held that judicial enforcement of Congress's inherent power was optional.

This power of Congress to punish contemptuous behavior itself was reinforced in 1934. In Jurney v. McCracken, the Supreme Court denied a writ of habeas corpus to a petitioner who had been taken into custody by the Senate sergeant- at-arms for allegedly destroying documents requested in a Senate subpoena.

The limitation on the president's pardon power was most comprehensively discussed in a 1925 opinion by Chief Justice (and former president) William Howard Taft in the case of Ex Parte Grossman.

_{Attribution: http://www.washingtonpost.com/wp-dyn/content/article/2007/07/20/AR2007072001802.html}

Pressuring those who prohibit officers from testifying by withholding salary

Thanks to Denis, who found this letter from rep. Mark Pocan to Secretary of State Pompeo:

According to this morning's New York Times story "Witness in Trump-Ukraine Matter Ordered Not to Speak in Impeachment Inquiry" published by Michael Schmidt and Nicholas Fandos, "The Trump administration directed a top American diplomat involved in its pressure campaign on Ukraine not to appear Tuesday morning for a scheduled interview in the House's impeachment inquiry." Who, Secretary Pompeo, prohibited the United States Ambassador to the European Union, Gordon Sondland, from appearing before the House Intelligence Committee today?

I ask that you direct the person who prohibited Ambassador Sondland from communicating with Congress to section 713 of Division D of Public Law 116-6 signed by President Trump earlier this year. As you are aware, this section prohibits paying the salary of any "officer or employee of the Federal Government who prohibits or prevents...any other officer or employee of the Federal Government from...communication or contact with any Member, committee, or subcommittee of the Congress." I believe the person prohibiting Ambassador Sondland from testifying before the House Intelligence Committee is in violation of this statute, and that their salary should be withheld until Ambassador Sondland appears before Congress.

Since Congress determines which offenses are considered abuses of office and qualify as impeachable, the withholding of witnesses could be considered obstruction of a Constitutionally authorized process that the President does not get to define or determine. Withholding witnesses could very well become actions he gets impeached for.

We consider this act today, and the withholding of the ambassador’s documents and other efforts to discourage other State Department witnesses, as further acts of obstruction of a co-equal branch of government.

-Adam Schiff

Washington Examiner: Schiff hints he'll push House to impeach Trump for blocking witnesses

Also, they can take it to the federal courts, which would add a significant delay, but this seems to be the most direct and immediate action at their disposal.

Trump could provoke more whistleblowers to come forward with damaging information. The people working for the executive branch swore an oath to serve the nation, not the person holding office. If they see the president continuing to flout the authority of Congress, they may simply decide that enough is enough and bring up yet more hidden information. If Trump blocks witnesses, he will end up provoking witnesses who are unblockable.

If this happens, members of the GOP may decide to run for cover instead of defending him. Rats will flee a sinking ship. Trump's GOP support is already declining. If this continues to happen, he may lose enough GOP senators that being convicted in the Senate becomes a real possibility.

House democratic leadership will have a serious fight on their hands. Their options, as I see them, in increasing order of radicalness* include:

Issue a Congressional Subpoena

News reports indicate that Ambassador Sondland had agreed to testify voluntarily but then was a no-show. Congress actually might get somewhere by using the formal process of a subpoena, but of course it is neither a punishment nor guarantee. News reports further indicate that a subpoena is in process.

Note however that President Trump (and his team) have already noted a flaw in house procedure. Without a vote of the whole house of representatives authorizing an impeachment investigation, a subpoena may not be valid! As Wikipedia explains the rule:

Congressional committee must meet three requirements for its subpoenas to be "legally sufficient." First, the committee's investigation of the broad subject area must be authorized by its chamber; second, the investigation must pursue "a valid legislative purpose" but does not need to involve legislation and does not need to specify the ultimate intent of Congress; and third, the specific inquiries must be pertinent to the subject matter area that has been authorized for investigation.

The Democratic congressional leadership has not called for such a vote, suggesting that many Democrats, particularly those in traditionally "red" states, may not want a recorded vote expressing their support for impeachment this before the election.

Contempt of Congress

Without a subpoena, it seems unlikely anyone is in contempt of congress yet, however if a congressional subpoena fails to produce a response, citation for Contempt of Congress is a possible follow-up. If a resolution of contempt is passed, there is a theoretical possibility that someone could be arrested by the Master at Arms, but in recent history that's hasn't happened very often.

Judicial remedies

Intra-branch disputes like this are nothing new. By filing a civil case in federal court, Congress may be able to press for an explanation or appearance. But normally courts don't involve themselves in "political questions", where no law clearly applies. Courts tend to leave these to the democratic process to resolve. Furthermore this bears some risks for the anything-but-Trump crowd: the broad concept of executive privilege probably applies, and a court ruling on a a fringe legal concept could backfire.

It might help to consider the question a case-by-case basis. Joseph Maguire, the acting director of national intelligence, did appear, so did the ICIG. In the news recently is Gordon D. Sondland, an ambassador, who did not appear. Questions over congressional testimony frequently don't get resolved quickly, and witnesses have a right to counsel, which takes some time.

The privilege issue may come with regard to separation of powers specified in the constitution. While the constitution doesn't mention executive privilege, courts have recognized it in the past, but not in every case.

The lower-conflict path to resolving it, on a case by case basis, would be to file a lawsuit in federal court and let the 3rd branch of government (the judiciary) sort it out.

Impeachment

Considering the particular case of State Department officials, the constitution clearly makes the conduct of foreign policy an executive branch function. Diplomatic functions are typically conducted with an strong expectation of privacy on each side. It would interfere with the conduct of foreign policy to testify in public, because foreign governments, Ambassadors, and even military allies would feel reluctant to speak frankly behind closed doors. For this reason, a rush to impeach based on diplomatic communications seems ill-advised.

*: The order in which these potential solutions are implemented seems to me to be a good indication of how far from center someone is.