Yes, appointments are still happening. They aren't taking calls but they have designated mobile lines where someone usually provides a text response within 24 hours. I texted my dogs name and reason for my visit to 9293574499 and his appointment was a week later.
I know Vet Wheels which is a mobile clinic does spaying/neutering on certain days of the week. You can get in line online. They have a website where you can see the schedule of their mobile clinic in different boroughs.
They are hard to get a hold of since covid hit. Usually there's a line of people outside Noone is allowed in except the pet. I know the aspca truck is functional for Bronx and Brooklyn resident with a low income its free with vaccines.
Thanks! Your answer is awaiting moderation.