Built-in Interpreters?

Is there a process by which someone can go to my website and just play an Inform game, without having to download an interpreter?

Yes–see §25.11. A playable web page in the Inform manual.

Or you can make a link to iplayif.com for Parchment. This is what the “Play Online” button that appears on IFDB does. You just put the url of the story file after “?story=”.
iplayif.com/?story=http%3A%2F%2F … LostPig.z8

I think there may also be a way to do the link thing with Quixe, but I’m not sure how you set it up.

iplayif.com includes a version of Quixe for Glulx games.