In my neck of the woods, the city recycling site sends me on a wild goose chase until I finally conclude that the only option is the appointment-only, Saturday-only hazardous waste drop-off at the dump. (There is a fee to recycle other stuff, so I have no doubt that medical waste is the same.)
The web page for that does NOT list medical waste or sharps containers, but the pages that direct me there do. Go figure. Anyway, I highly resent the runaround, and I much prefer to do my business on Fridays, when I am actually out running errands. So I stack up the containers at home and will continue to do so until desperation sets in.