Yes, good question. There must be some kind of way of doing it as my Dentist confirmed my appointment for next month, about 10 mins after I had made, and I can't see the Dental nurse fiddling around with a Mobile phone to do the confirmation.
there used to be a few sites which offered free texting, one was blue(something) or (something)blue.
Google 'free text message' and see who shows up.
Mamya's links are 99% helpful so try hers first.
Mikey, our local taxi firm does that sort of thing too, their automated system cost them the £300,000 so it's eye-watering to think what it cost the NHS