We will be traveling / backpacking in France for several months. We'd like to send packages from hotel to hotel when we are in towns, with supplies we don't want to carry. Town stops will be 5+ days apart. In the US we have UPS pick up at one stop and deliver to the next., so we don't have to worry about finding a post office or its opening hours. Is there a way to do something similar in France?
I'm afraid I can't help with the shipping question, but want to be sure you realize that "several months" needs to be less than 90 days (including both arrival and departure days), due to the Schengen limit.
If you do that it might be wise to warn the hotel ahead of time lest they consider that that package for you in the room just behind reception may be a security risk.