I've used that site many times myself.
Some starter Pokemon aren't listed there possibly to prevent people from recruiting starter-only Pokemon early in the game. Or, the people were a bit too lazy to add Shinx on there. It does say it's a beta and it'll probably never get updated. It sucks, but that's how it is. If the Wondermail S generator doesn't have it, then the only way is to recruit one in-game by first beating the main storyline, then going to Amp Plains and recruiting a Shinx by defeating it (or multiple of them should the first time not work). They're not that hard to recruit, though.
The starter Pokemon don't appear in dungeons until after the main story is beaten. Unless someone finds a recruit Shinx wondermail in-game on the bulletin board or outlaw board and then gives you the wondermail code for it, then you'll unfortunately be stuck with just simply recruiting it yourself. It sucks and probably time consuming, though it's worth the time. Good luck to you.
My own knowledge and experience. Explorers of Sky is fun, and I have a copy myself.