How do you use tabletop simulator? Just looking at the description, it seems like you'd basically have to program the whole Dominion game into it and then add the fan cards.
Tabletop Simulator is a physics sandbox. It means that you, and whoever you conne t to online, can use their mouse to click on stuff, drag it around and let it fall down somewhere else. No programming involved at all. It's like on a real table - once you have the cards, you can either use them to play a game, or just throw them at your opponent, or whatever you can do with a physical card. Rules are not part of TS, you do that on your own. The software is more than basic physics, though. It also comes with a lot of prefabs and quite a few given functions, such as cards. You can deal cards, shuffle cards, and put them in your "hand" - an area where only you can see their front side.
About how to get cards in there, most cards exist allready as free workshop items. The only thing you have to do is download those. To add your own, create a file containing your fan cards (there's a software for that that goes with TS, allthough a little hidden), upload that template on an image hosting site like imgur, and then import it. Takes five minutes at most.
TS has no concept of rules, and is a bit slower than Dominion online, so i hope this doesn't discourage people of using the official implementation. The use case for dominion is really mostly fan cards, but it's very useful for that.