How do I program each button to not only light up the led, but also push different data to the form?
You can't program the buttons. Put them back on the shirt you cut them off of, and plan to use switches, instead.
You can program the Arduino to react differently when each switch is pressed.
I just cannot figure out how to insert the php
Insert it where? That php script is going to do the same thing every time it is invoked. Hardly seems like the kind of thing that you'd want to call when 6 different events occur.