Basically, no. Unless there is some gross discrepancy (M14s vs flintlocks), small arms make little difference to the final outcome, though they may decide local engagements.
Discipline; training; experience; support, both logistic and combat; morale; leadership; belief in the cause; all are more important than the type of small arms employed. Germany and Japan did not lose WWII because we had the M1 rifle and they had bolt actions; they lost because we had a vastly greater production and logistic capability, and massive air power that destroyed factories and supply channels. (For that matter, England was on the winning side, and THEY had bolt actions.)
The place small arms make a difference is in the morale of the soldier. The individual weapon is the soldier's friend. Soldiers have been quoted as saying, in effect, "The other stuff is the government's, this rifle is mine".
If the other side has small arms with some real or imagined superiority, the soldier soon begins to wonder why his nation cannot provide him with a better weapon. And that is where morale begins to break down and the loser psychology takes over. If this continues, and spreads into other areas, the war is lost, no matter how objectively "good" or "bad" the soldier's personal weapons are.
Jim