Note: This took me a lot of trial and error, and I averaged about 1/10 matchmade games being able to quit and then go into another game through navigation, it is frustrating, and entirely dependant (as far as I know) on if there are any new matchmade games that you can join as stalker (which also fit your region and ping settings).
Guide:
When clicking shared purpose in the ordis tab on mercury, be ready to open up your menu immediately, as you will be attempting to stop the matchmaking manually.
When you get put into a team and are still in the relay, with the match start timer counting down, attempt to press the x (cancel) option in the top left/center part of your screen. This will stop you and your team from going into a game.
Now, before your team tries to restart matchmaking somehow, go into the closest navigation terminals nearby and select the mission you would like to play, you'll know you will load in as stalker if it says looking for targets, then targets found, instead of looking for players.
Trivia:
Adjusting your matchmaking settings (Public, Invite-only, solo) does not seem to affect your matchmaking as stalker, and you will still be paired with a squad.
When in System settings, turning off UPnP will not affect matchmaking, however turning off NAT - PMP will prevent you from pairing with any squads.
If you are able to spawn a kuva lich as stalker, there is a possibility that it will have the model that has your warframes head model on the side of it, if you mercy killed it as stalker, it will have stalkers head on it I believe.
The progenitor element for a lich killed by stalker will be ???
When in a survival mission or one with a timer counting down, the player using stalker will imediately exit the mission without any choice.
Things to experiment with:
Ways to dodge the que and hopefully get into a non-ascension match at a better rate, try using friend invites, network filtering, or other methods to give the stalker player as much control over matchmaking as possible.
Any gamemodes that break with Stalker in them (endless timers, solo games despite forced matchmaking)
Thank you and please post any more facts in the comments!