I know how you can send text messages through PHP like you send an email (explenation: Sending a Text Message (SMS) with PHP and more info: How to Send Free Text Messages Using PHP) but the person has to give their cell phone provider (ex: Sprint, AT&T, Verizon). I want to know if there is a way to have people input their cell phone number for a text to be sent without requiring their provider.
I don't believe that you will be able to do this without signing up with an SMS Gateway service who will do it for you (at a cost). You might be able to hook up your PC to a cell phone and have the cell phone act as your gateway but I don't know for sure that you can do this (or how).
I'd like to believe that there is an intended level of secrecy around this to avoid spam sms messages being sent too easily. There are probably financial reasons as well.
So the way it works for big companies like Google who allow you to send texts form Gmail and stuff is they have a gateway which is (in simplest terms) a phone connected ot a computer that sends the texts?
They are big enough that they probably have their own gateway. The gateways are probably done by an internet link to one of the cell providers and they probably do the rest. On a small scale, you might be able to do it through a phone but on a large scale that wouldn't be practical.