Yes a 2 stoke will run on gas, but as you asked, how to lubricate the engine while its running? Just guessing here, but you would need a way to spray oil into the intake. The oil could be very light weight, maybe even light enough to use the existing carburettor? It would need to be jetted to suit, but I think it could be done, the carb would meter in oil depending on throttle position and rpm.
Mind you, a lot of work to get it running right, would be easier to find a 4 stroke engine, a lot easier to convert to gas.
Glenn