I am playing a Beast Master ranger. My baboon animal companion* obeys commands as best as it can. Can it use a wand of magic missiles?

The baboon has hands, and the wand does not require attunement nor a command word to be spoken.

RAW, maybe (but probably not)

The section you quoted goes on to say:

You can use your action to verbally command it to take the Attack, Dash, Disengage, Dodge, or Help action

It is not entirely clear from context if this is supposed to be an exclusive list or not. If we are to take this as an exclusive list, then the answer is no, because using a magic wand is not on that list - but doing so would have other somewhat weird consequences, like your Baboon being unable to open doors (Use An Object), which I have a hard time imagining is an intended consequence of the description; On the other hand, if the game gives you a list of things and doesn't explicitly say that they are meant to be an examples they are usually meant to be taken as exclusive.

I'd probably allow it, however you should note that commanding your baboon to do this would require your action at any rate, as per above + the 5e errata:

If you don’t issue a command, the beast takes the Dodge action

Rules As Written, A beast companion can only activate a magic item during combat if you are incapacitated and it may require them to have focused on the item for an hour at some earlier point in time.

First some collected evidence:

From the question "Can a familiar attune and wear or use a magic item?" we learn that there are no rules forbidding any creatures from wearing, using, or attuning to any magic items which is also corroborated in the question "Can beasts benefit from magic items?" where it is stated that:

All creatures can benefit from and use magic items subject to their anatomical limitations and other capacities.

Additionally the question "Can an animal attune to a magic item?" explains that any animal (and so certainly beast companions) can attune to a magic item so long as they focus on the item for an hour. Upon attuning to the item:

[...] the creature gains an intuitive understanding of how to activate any magical properties of the item, including any necessary command words.

The section on "Identifying a Magic Item" also states:

[...] A character can focus on one magic item during a short rest, while being in physical contact with the item. At the end of the rest, the character learns the item’s properties, as well as how to use them.

The fact that you can attune to an item even if it doesn't require attunement is explained in the question "Does a wand user know how many charges remain?"

Thus a beast companion will know how to activate a magic item and has the capability to do so, but the beast companion's description states:

The beast obeys your commands as best as it can. It takes its turn on your initiative. On your turn, you can verbally command the beast where to move (no action required by you). You can use your action to verbally command it to take the Attack, Dash, Disengage, or Help action. If you don't issue a command, the beast takes the Dodge action.

That list of actions comprises the only set of actions you can command it to take during combat. And if you don't issue a command, it simply takes the Dodge action. Thus in-combat you could not command the beast companion to use a magic item.

That said, the beast companion section goes on to state:

If you are incapacitated or absent, your beast companion acts on its own, focusing on protecting you and itself.

In this case, the companion acts entirely on its own, and thus is not restricted by the command rules. This would allow it to take other actions, such as Hide, or Use and Object, or using any magic object.

I agree that this is nearly nonsensical that a beast companion cannot Hide nor Use an Object during combat unless its master in Incapacitated, but that does seem to be what the rules state. I've now asked whether this is the case in the question "Does a Beastmaster Ranger's animal companion gain more actions when its master becomes incapacitated?".