The Animate Objects spell states (emphasis mine):
An animated object can be of any non-magical material. You may animate one Small or smaller object or a corresponding number of larger objects as follows: A Medium object counts as two Small or smaller objects, a Large object as four, a Huge object as eight, a Gargantuan object as 16, and a Colossal object as 32. You can change the designated target or targets as a move action, as if directing an active spell.
So enchanted objects are not a valid target.
Also, a construct could be custom-built to hold a Wand but it would not be able to use a Wand.
Wands are activated by using Spell Trigger:
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.
And to be able to use Spell Trigger, you need to have the spell on your class list:
Spell Trigger: 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.
Animated Objects don't have any class levels.