1. This year, what do they plan to visit?

They plan to visit nursing homes in the town.

2. This year, what place do they plan to visit?
They plan to visit nursing homes in the town.

3. This year, where do they plan to visit?
They plan to visit nursing homes in the town.

(Which questions are grammatical? Are they all acceptable?)

All three questions are grammatical and acceptable. They only vary slightly in terms of syntax and word choice, but they convey the same meaning and intention.