If you can get regular Internet access, I would be surprised to hear that the app isn't able to connect.
These errors display when the repo URLs can't be reached:
https://gitlab.com/4TU/hb-appstore/-/blob/master/gui/MainDisplay.cpp#L63-82
The FAQ page probably isn't too useful, basically suggests trying a connection test:
https://faq.fortheusers.org/#says-no-internet-is-present
I would suggest checking if the web version at
https://apps.fortheusers.org loads properly on the same network and DNS server.
You can also try entering recovery mode, by pressing L/R repeatedly while the hb-appstore launches. This mode has more generous error handling:
https://gitlab.com/4TU/hb-appstore/-/blob/master/console/Menu.cpp#L166-171
Recovery mode also has a "Reset data" option which may help if the repos file is corrupted, but it's probably overkill to try this option, instead of just checking the repos file. This file is located at sd:/switch/appstore/.get/repos.json and should be pointed at
https://switchbru.com/appstore
. Deleting this file causes it to be re-generated on the next launch.