"Downloaded Vehicles", do you mean "subscribed in Steam" or "downloaded from a website" ? All vehicles subscribed in Steam will be added automatically and the game will load these. If not loaded by the game, they would not show in the game.

Here is some solutions options to determine, if it has been loaded:
1. Install Loading Screen Mod and activate the Reporting options to see, if the Game loads your vehicles.
2. Activate the logging in AVO and check the Output.txt-logfile, if there is any error message. The output.txt logfile can be found via AVO mod options and clicking to the button in the support section.

One solution could be unsubscribe/subscribe the assets again to enforce Steam to delete and re-install the vehicle mods.

AVO only shows, what has been loaded by the game. Maybe there is some other conflict with your installation, as you say it works on a different computer.

Thank you for your help, it works now! I checked the Loading Launcher Mod which I had already installed. There was a unchecked box which said "Load custom assets". Must have slipped my eyes. I spent hours searching for a fix, and in the end it was such an easy solution.
(Yes, I meant subscribed in Steam.)

Perfect, glad that we found the setting so quickly, else it would have been a hard job to track it down...