I’m having the exact same problem described here, The upload media in the local (vagrant:homestead) wordpress library is not working.
I’m using MAMP for Mac and set Bedrock up as described in the docs, and using the default 2021 theme as well. When I try to upload to the “Media Library” via the back-end, I get an error msg stating “Something went wrong. Try again later.” and then I get logged out - bizarre.
I set up a normal WordPress install and everything works grand on it.
P.S. I have both installs in the same place, and as far as I can see there is no difference in permissions.