For what it's worth, I didn't enter an email when I made my account on Beehaw and I didn't have any issue with getting approved. I've seen the admins saying that there's a pretty heavy backlog of new accounts to approve at the moment so it could be that they just haven't gotten to you yet.
Apparently there's a known bug where if your chosen username is already in the instance the page will just hang and the admins will be none the wiser that you tried to sign up at all, so if you find that you haven't been approved after a few days it may be worth trying again with a different username. Source here: https://beehaw.org/post/562922