This is a tradition that's becoming more and more rare, but yes they are allowed to come to you and ask for money. You have no obligation to give them any.
I don't know why it's only the trash collectors but my guess is that it's from a time when housewives were a thing and they would get to know their local mailman, trash collector, milk delivery guy, and would give them a small stipend at the end of the year.
Here's a little extra info from the RTBF