How do seniors in care facilities vote in elections?

By absentee ballot.

And just a warning, make sure no League of Women Voters person has come around to re-register your relative. When my mother was first in her nursing home, she was a mile away from her house, but in another city. When the first local election came up after she moved, she somehow received a ballot for the city where the nursing home was rather than where her home was located. While it didn't matter a great deal because she was not competent to vote any longer anyway, someone had gone to her room (I assume) and asked her if she wanted to register to vote. That was completely inappropriate in my opinion, as her visitors should not have been strangers, especially as she had dementia. I was livid, and no one at her place knew who would have been in her room.

Absentee ballot.
Or if a friend or relative picks them up they can vote in person.
A very large facility may actually have a voting site on the property.

Absentee ballot, with or without help, as needed. Just like if you were disabled at home. Each state has its own laws for absentee ballots. In my state all voting is done by mail with voter's pamphlet and ballot delivered by the post office. Your ballot can be dropped off at a ballot box or returned by mail, postage paid.