After selecting the picture you want as your forum avatar, there should normally be a save button at the bottom of the page to complete the process. If you are having issues scrolling down to reach the save button, please try logging in through a different web browser. Once logged in under a different browser, see if you now have the save option so you can change your forum avatar.
Here's what I did, @Kruemeli. Select the avatar you want, then click within the window pane, but not on any of the pictures (I clicked on the space to the right side near the scroll bar). Press tab, then enter.
I think that tabs to the Save button and triggers it, because for me that reloaded the page and I had my selected avatar.