Is it appropriate to contact a professor about paid summer research before beginning PhD?
Yes, you can ask. If the professor has grant funds and a need for someone to fill a slot then you might be successful. It needn't be the person you intend as your advisor, and if you are entering with just a BS then it is unlikely that you would choose a final advisor so soon.
But asking is not a problem.
However, you might also consider that taking a bit of time off might be better overall, to avoid burnout that can arise after an intense degree program. If the money is essential, however, don't hesitate to say that.