So I was GMing for the first time my custom Pathfinder campaign, and my Rogue uses a wand of magic missile with 8 charges left in it against the last opponent of the session, and vaporizes him. I instinctually said he couldn't do that, and he explained that he only needed to make his Use Magic Device (UMD) check to use the wand (which he succeeded) and he's using the wand, not the spell, so he can use multiple charges in the same turn with the same check. It was the end of the session so I didn't want to bog us down with an hour long rule check, so I allowed it and my villain was vaporized by the remaining 8 charges (we are low level still). I've been searching for a ruling on this all afternoon yesterday and this morning, and just cannot find a definitive answer.

My reasoning is:
Activating a wand

Wands use the spell trigger activation method, so casting a spell from
a wand is usually a standard action that doesn’t provoke attacks of
opportunity.

(We are in agreement here)

Spell Trigger Description:

Spell trigger activation is similar to spell completion, but it’s even
simpler. No gestures or spell finishing is needed, just a special
knowledge of spellcasting that an appropriate character would know,
and a single word that must be spoken. Spell trigger items can be used
by anyone whose class can cast the corresponding spell. This is the
case even for a character who can’t actually cast spells, such as a
3rd-level paladin. The user must still determine what spell is stored
in the item before she can activate it. Activating a spell trigger
item is a standard action and does not provoke attacks of opportunity.

(He still agrees with me up to this point)

UMD Description

Use a Wand, Staff, or Other Spell Trigger Item: Normally, to use a
wand, you must have the wand’s spell on your class spell list. This
use of the skill allows you to use a wand as if you had a particular
spell on your class spell list. Failing the roll does not expend a
charge.

So magic missile is now as if it were on his class list. Magic Missile has a casting time of "1 standard action"

My understanding here is, is that you can only cast once per standard action. It would be the same in my mind, a wizard preparing magic missile 3x in one day, and expending them all in one turn just because he channeled the magic missile.

Neither of us can find anywhere it say that only 1 wand charge per round, and that is what I think his hangup is.

What stops someone from making a wand of magic missile and use all 50 charges from one turn on the next boss we fight?

You cast one instance of a spell at a time from a wand.

From the rules for wands:

Activation: Wands use the spell trigger activation method, so casting a spell from a wand is usually a standard action that doesn’t provoke attacks of opportunity. (If the spell being cast has a longer casting time than 1 action, however, it takes that long to cast the spell from a wand.) To activate a wand, a character must hold it in hand (or whatever passes for a hand, for non-humanoid creatures) and point it in the general direction of the target or area. A wand may be used while grappling or while swallowed whole.

When you use the wand to cast, you are casting a single instance of a spell, just as you are when you cast as a spellcaster. You cannot use up multiple charges at once to cast multiple instances of a spell.

Since you use 1 charge per instance of a spell you cast from a wand, and it takes at least a standard action to cast a spell from a wand (per the above rules), you can only use 1 charge a turn from a wand.†

You most certainly can use more than one charge per round when using magic items. Some effects of the staves even require you to so. You can't find a limit in the rules because it is not here.

The reason why you cannot fire more than one missile from a wands per action is different. Per your own quote:

Wands use the spell trigger activation method, so casting a spell from a wand is usually a standard action that doesn’t provoke attacks of opportunity.

Emphasis mine. Even if you use UMD, you are still casting a spell, so all of the usual limits of casting spells apply. And you can't combine three identical castings of a spell into one action, casting a spell (singular) takes an action. This was already discussed on this site recently.

Rule of fun and what would be the ruling at my table

UMD is all about doing impossible things with magic items. Let him do this, but pump up the difficulty, make it use 3d4 charges instead of 3, and make mishap easier. If he still wants to try, yeah, I'd totally agree. But it has nothing to do with the rules as written.