My witch is almost at 7th level. She has a boon that will allow her to take a Soul bound doll as an improved familiar. Can the soul bound doll use her wands since the witch gets her spells from her familiar anyway?
As in without using UMD to do so? No. The familiar is merely a carrier of the spells, and has no access to them.
Since you said you have a Boon, I am going to assume you mean for pathfinder society. In pathfinder society, only certain familiars can use wands.
Specifically, under the PFS Faq it states:
It is intended that animal companions or familiars can not activate magic items.
Wands would fall under this category. However there is a specific post from a former campaign coordinator Mike Brock (which is pfs binding) that states:
Wand use doesn't require the book for those few improved familiars - brownie, faerie dragon, imp, lyrakien azata, mephit, quasit, sprite familiars - gained with the Improved Familiar feat
It can use a wand since it is slotless. It uses its master's UMD. No other animal companions or familiars can activate a magic item.
So some familiars can use wands, that one cannot. As to the question of whether it could use it without umd, that would also be a no.