I’ve now found that through the /e/ apps store I can install and update most apps. It is a two step process though. First I try to install / update the app in the /e/ store app. Most often this leads to a download of the desired app, but not an installation or update. It looks to me like once the file has downloaded, the /e/ store app terminates. If you go back into the /e/ store app it will look like nothing has happened. In my downloads folder though, I find a new downloaded file. Mostly this file will be named appropriately (e.g. ‘Signal Private Messenger’) and have the type ‘Android Application’. If it has the type ‘file’, you’re out of luck. If it has the ‘Android Application’ type though, by tapping on it, and then opening with the package installer, this should install / update the desired app without any further problems.
You can then check again in the /e/ store app to see if there is still an update available, but hopefully you will not see the app in updates, or if you can, there is only an option to open it rather than update it.
I’m still running /e/OS 0.9-2020061158052 on a Fairphone 3.