You can try Grandma & Grandpa'a (Scott) at (352) 552-8478 or Jim Followell at (352) 259-7525. Perhaps they'll have something available.
And yes, The Villages Golf Car store rents carts as well.
Best wishes - Jan. through Mar. is a tough time to get a rental.
However, given that you're wanting it for the whole month will be an advantage.
Bill