This is going to sound like an odd question, but here goes.
I am currently a customer of AT&T. The Verizon store has a ringtone that I cannot find anywhere else. If I download this ringtone to my AT&T phone, will the ringtone successfully install and play on my phone? Or will it only work on a Verizon phone?
You cannot purchase ringtones from Verizon to use on AT&T's phones. You should switch to Verizon. Then your problem will be solved. There are also many ringtone maker applications and such available you can make it for yourself. That's what I do.