The Chelsea Health Center, located at 303 9th Ave. between 27th and 28th St., is giving flu shots until the end of the month. They're open from 8:30am to 2:30pm, Monday through Friday.
I got this information from the Department of Health (212 676-2273). When I asked if it was too late in the season to bother with a shot, the woman responded that since the shots are being offered, it means it's a good thing to get one.
UPDATE: As it turns out, we DO live in a communist society. The shot is free!