I saw this quote on page 36 of the April, 1945 issue of American Rifleman (talking about club .22 caliber match activities):
"Even the critical ammunition shortage did not slow down activities to any extent. Each club member was issued a ration card which entitled him to fifty rounds every two weeks. Team members received a "supplemental" ration of 25 extra rounds for each match."
So it sounds like some small ration of .22 ammo was available to club members for competition.

Peters, Remington and Winchester ammunition ads talk about "more" or "unlimited" quantities of .22s being available when peace comes at the end of the war. Remington's December 1945 advertisement mentions that target shooters have been waiting for Remington target rifles and match ammunition.

I don't know about centerfire ammo availability but it seems that plant guards and police were encouraged to use club ranges for training.