Can I split a 120v outlet off a 240v air conditioner circuit?
No. The National Electrical Code does not allow line to neutral loads on a 240 volt line.

Besides that the line to the air conditioner will not have a neutral for the 120 volt line anyway. If you cheat and use the equipment ground that is another very dangerous code violation.

Just run a new wire. It is not worth you and your family's safety to save a few dollars.

Good luck and stay safe!