In Munich, you definitely need a permit - the city has strict rules about this. You're allowed to rent out your apartment for a maximum of 60 days per year on a short-term basis without having to register it as a lodging business. Anything beyond that requires a permit, and you'll need what's called a "change of use permit" from the city. The problem is: Munich barely issues these permits anymore because the housing market is already tight. So you should first contact the District Administrative Office (KVR) to find out whether a permit for your situation is even possible.
Here's how I'd approach it concretely: Check the City of Munich website for the current requirements on change of use - the rules keep changing. Then contact the relevant department and explain your situation. They'll tell you whether you have any chance at all or if the 60 days per year is the maximum for you. If you don't register and rent it out anyway, you risk fines in the four-digit range.
Alternatively, you could rent the apartment long-term - that's completely unproblematic and covers your utilities too. Just not as flexible as Airbnb.