Hmm... that's difficult! I can only think of something that's a) open from the top or bottom and b) pushes the bullet towards the barrel in one action, Let's see what I'll be able to come up with. It seems like a simple mechanical design question to me.
Today is a good day! Atleast when judging from how I woke up today. However, I already found and fixed a mistake when trying to fire the bullets. I kind of twisted the rubberbands wrong and now I'm having a pretty responsive and qickly triggering Lego gun. Of course, the magazine is still sloppy, but I'll change that. I thought about how one could force the bullet to stay in one place and I think I found a good solution for that. It helped to experiment with slide-out mechanics - this can be used to lengthen the chamber itself on several occasions and then shortening it back. So the idea is to make a chamber which is shorter by default, stretchable and thus able to push bullets towards the lock. The only action I can think of that'd useful there is break-action. It allows the bullets to pop out will push it into the chamber when closing the breech. But uhhh, it's probably not so nice to chamber with a bullet potentially blocking while rotating the chambers.