Is there any way to enable some users to manually award achievements, ideally from within a community page, without giving them admin access?

Theoretically yes, we do expose Apis to award achievements, but awarding one requires a high permission level( as in manage membership permissions, the permission that allows all users to be edited/created) so you can write the code, execute it as a privileged member and it will work but if you are not really careful you could expose security issues, especially since you would need logic to limit what achievements, what users, etc.