I have found that App Lounge behaves a bit as if the network is too busy to fully complete tasks sometimes.
I would start by clearing storage within App Lounge
Settings > Apps and notifications > See all … apps > App Lounge > Storage & cache > Clear storage | Clear cache
Now try to install one app at a time. Should you get a stall situation as you report I would not carry on and try another app, but rather come back and try the initial app later.
Also Advanced Privacy is known to interfere with App Lounge. Please test App Lounge with Advanced Privacy / Hide my IP turned off.
A clue to the “too busy” situation on the server is what I call an animated blank screen.
If on starting App Lounge you get an animated blank screen lasting over 15 seconds, please treat this as a “Please try later” condition.
In the next pair of images I search Fennec. I have found that it is significant that you move on the the second screen, by selecting Fennec from the initial “Search” screen, as the behaviour you report seems to me more likely when you attempt to hit “INSTALL” from “Search” rather than “Install” from the target app’s “Own screen”