Hello Florian
Rene has a lesson dedicated to this very question. You can find it here.
However, to provide a quick answer to your questions:
Yes it should work. In Rene’s lesson, he provides to solutions to making multicast work without a router. One is to make one switch a querier. The other is to configure PIM on the SVI interface of one of the switches. Take a look at the lesson for more details.
Essentially, yes, this is one of the fundamental functions of a multicast router.
I hope this has been helpful!
Laz