Bought a new 7 speed sprocket
 

for my old Specialized Hard Rock and a removal tool. The removal tool was the wrong size (I thought it was supposed to fit all shimano casettes) but I managed to get the old cassette off with my home made chain whip and a hammer. Once I removed the old sprockets, turns out my new ones were the wrong size. Is it possible to get a new hub that will fit my new sprocket or should I just get an entire new rear wheel?

sounds like you dissembled a freewheel

I think so. Problem is the new one doesn't have a big enough hole. Won't fit on the hub.

http://bicycletutor.com/replace-freewheel/

Cassettes and freewheels are two different animals

There are two sizes of shimano splined freewheels. There's the big size, which is by far the most common, and a smaller splinned size that shimano used briefly in the 80s. if you've got the later, take the wheel to a bike shop that's been around 20 years or more, and have them pull it. Unless your a vintage freak, you may never see another, and it's not worth trying to buy the tool.