We love that our ECV user can get transported too!
You use the Lyft app and select Minnie Van on the scroll of car options. However, they only run on Disney property and to Four Seasons, Swolphin and Shades of Green. The Minnie Van airport service is not booked on the app so you most likely can't use a Lyft gift card, also I think it's still only available to CL/VIP guests at this time (it's not cheap at $200/one way). I've never used the airport service, I like it specifically for Swolphin-MK and DS (I have the worst bus luck there).