Howdy,
I did something like this about a year ago with ...
http://www.clickatell.com/
If it's pretty much the same now you just sign up, get their API and your off and running with their service. Just setup a form that dumps your phone number and any other info the API calls for and dump it into a script that sends the connection link.
They charge you by each message. I think they give you ten test messages. When I was working with this a lot of the downside was getting calls back to USA through certain carriers. Lot's of politics and red tape. Maybe things have changed...
Just google sms gateways.... should find a bunch.