The French OTL had longbowmen, and plenty of them. This included the "Franc archers", a sort of reserve army where each parish was expected to recruit and equip an archer, as well as large numbers of mounted archers in the standing ordonnance companies. These longbowmen actually outnumbered the men-at-arms. However, this was only in the latest stages of the war.
Before that, they had 4,000 archers and 1,500 crossbowmen in the vanguard at Agincourt, though they were ineffective due to being deployed behind the main body, and their lack of armour.
They also made extensive use of Genoese and Spanish crossbow mercenaries, who if equipped with pavises to cover them while they reloaded could have been much more effective. At Crecy, where they were infamously slaughtered, they were forced to advance with wet strings and no pavises.
Plenty of archers, both English and French, had horses and could serve as light cavalry if need be. Indeed, the French mounted archers were often minor nobles who couldn't afford the extensive equipment of a full man-at-arms.
Basically, what you need to change with the hundred years war French is not the equipment, but the tactics. They had a very formidable military system that was often used in a moronic way.