Does mage armor work with the druid's wild shape? A player might multi-class for that. As a whim I feel it's OP but can't find anywhere that it's not allowed.
Yes, Mage Armor continues to work as long as your beast form is unarmored (no barding!). This doesn't 'stack' with whatever AC the beast form has, however- you pick either the beast's normal AC, or the 13 + Dexterity modifier from the Mage Armor. See Basic Rules p.9:
Some spells and class features give you a different way to calculate your AC. If you have multiple features that give you different ways to calculate your AC, you choose which one to use.
Do keep in mind at the time of casting the person benefiting from Mage Armor has to be unarmored or else they are not a legal target for the spell, and that the spell ends if the target becomes armored. If the druid is not yet in beast form and is wearing leather armor, they will have to doff it before getting the spell cast on them and cannot put it back on unless they want the spell to end.
In addition, if the spell is cast upon them while they are in beast form, it will end when they shift out if they are still wearing armor in their humanoid form.
Also worth noting is that transforming into a form with natural armor does not cancel out Mage Armor, either. From SRD p.255's description of monster AC:
A monster that wears armor or carries a shield has an Armor Class (AC) that takes its armor, shield, and Dexterity into account. Otherwise, a monster’s AC is based on its Dexterity modifier and natural armor, if any. If a monster has natural armor, wears armor, or carries a shield, this is noted in parentheses after its AC value.
Note the distinction in referring to worn armor vs natural armor; I believe this and the language of Mage Armor specifically referring to putting on/wearing armor is sufficient to indicate natural armor does not prevent Mage Armor from functioning.