Rule to add member based on name in checklist

I am looking for a way to generate a rule in Trello that will pull any name that is typed using @ in a checklist and apply that individual as a member of the card automatically

There isn’t a way for you to intercept that as the user is typing. You can get an updateChecklist action sent on a webhook and use that as a trigger to go figure out which user was added to the checklist and then add them to the card.

Doing it this way does require managing webhooks: Webhooks