Yes, it can use the wand
The creature animated by tiny servant is a creature and the spell places no restrictions on what kind of actions it can take, or be commanded to take.
As a bonus action, you can mentally command the creature if it is within 120 feet of you. [...] You decide what action the creature will take and where it will move during its next turn [...]
Therefore, it is able to use any general magic items, including a wand of magic missiles.
For other magic items, a tiny servant can even attune to (up to three) magic items, but not ones that have requirements (such as being a spellcaster) that it does not fulfil (such as the wand of fireballs). It is also limited by its carrying capacity — which for a Tiny creature with a Strength of 4 is 30 lb — but this is not a problem for most items.