There are two time periods in question here: before and during the Battle of France, and afterwards with the Free French Air Forces. Before and during the Battle of France, the French did have their own air force. Wikipedia's description will suffice. Combined, the Allies had a numerical advantage in nearly every category. The most modern French fighter design could (and did) compete with the Bf-109 on relatively equal terms, and the French did reasonably well in the air. The Germans did have a higher sortie rate and were thereby able to win air superiority at key moments. This was a gamble by the Germans, because it caused higher wear on their men and machines, while the French were holding some strength in reserve for a longer conflict. As it worked out the battle was over in weeks, not months, so the German gamble paid off.

The Free French also had some air units. They flew a mix of survivng French, British, American, and other designs. This list shows all of the different types flown, but not the numbers of each.

For an examination of the performance of the French air units in the Battle of France, The Age of Airpower by Martin Van Creveld has a good summary (among many other books). For relative aircraft production Van Creveld has a good overview, but Joseph Maiolo discusses the difficulties faced by the French in more depth in Cry Havoc.