Earlier this week I heard of two people who had decided to take a last minute break at a friend's villa in the south of France arrived at Nice airport hoping to rent a car. They could not find anything apart from an Audi TT that was rather expensive for the week due to the type of car it is.
One of their friends contacted several car rental companies on the phone and was politely told they had no vehicles available.
Eventually they were able to source a vehicle (but would not be able to collect it for a couple of days) from a vehicle body repair company that provides courtesy cars. This was only possible because the villa owner actually knew of this company.
I normally use Holiday Autos and looked on their website seaching for cars at Nice airport for the next day. There was a message saying "We do not have any cars that meet your requirements" and I found the same situation on the 11th July looking for a car from 12th.