Can you do paid external work under a PhD contract in France? If someone has a PhD contract in France and wants to work during summer for instance, is it fine if that person receives any external support during that time, outside the PhD contract support?
It depends on the work you plan to do (see here, paragraph IX). 


You are NOT allowed to do extra activities of the kind that could be part of your PhD contract (teaching, diffusing scientific information, expertise...). If you want to do this kind of work, it needs to be under your PhD contract.
For the other cases, you need to obtain an authorization from your employer. If your employer is the university, they will ask your advisor's opinion and evaluate whether the extra work risks compromizing your PhD before making a decision. I don't know if such authorizations are commonplace.