Can a level 15 player join a Normal mushroom battle?
Yes. A level 15 player can field up to 25 Pikmin, and a Normal mushroom only needs a squad of 10 to start — so you can join.
Your maximum squad size grows with your level, and each mushroom size has a minimum squad you must field to start a battle. If your cap is below that minimum, you can't join — no matter how strong your Pikmin are.