No, the tank and pump are seperate, unless someone modified it. A water line is connected at the pump, but there should be a back flow preventer that keeps city water from flowing into the tank. Some people will add a valve so they can fill their tank from the internal plumbing, so check for this mod.
If your tank is cracked it might indicate it has been frozen, therefore other plumbing may be suspect. If you have not already, try hooking up to water to pressurize the fresh water system, and fill up the gray and black tanks to see if you have any leaks before startring out.