You can walk around the town of Jackson.
The tram is in Teton Village. There is a bus service to and from Teton Village -
Or you take a taxi, Uber/ Lyft or rent a car.
To visit the GTNP and the rest of Jackson Hole, I'd hire a private driver or just wait to see if on the tour.
The Lexington is in the town of Jackson, WY. Jackson Hole Mountain Resort is at Teton Village, WY about 12 miles from Jackson and as noted by the nice lady reachable by bus. The tram will not be running for several weeks starting in early October.
The bus in the town of Jackson ["Town shuttle"] is free.
The Lexington is less than two blocks North of the Town Square.
Thanks very much onemoreroad and Absarokanaut. Your info are very useful. Appreciate very much.