If I have my current phone plan through Sprint [or whomever], is there a reason that I shouldn’t get an iPad through them?
No, no reason at all, if you have been happy with their connection.
By the same token, the iPad is a separate bill, so you have no reason or obligation to go with the same carrier on the iPad as you have on your phone. In San Antonio, I generally recommend AT&T, but y’know, they’re all bloody bastards who have finally caught up to semi-decent technology. There are only fine, subjective distinctions between any of them.