Will a submersible pump run without a ground wire?
A submersible pump will run without a ground wire, the primary function of the ground wire is operator safety. Running a pump without a ground wire puts people and property at risk. Not only is it not a good idea, it is also dangerous and often in violation of regulations, and you could be held responsible for damages. The US National Electrical Code and Canadian Standards Association recognized that submersible motors are not always in steel cased water wells. With the on-set of non-metallic materials in pumps, drop pipes and well casing, code standards require running a ground wire from the motor to the electrical power source ground, IE breaker panel, connecting to all metal parts along the way.
Will a submersible pump run without a ground wire? Yes. But for safety DON’T DO IT. Bring it up to code.