Do any of the beaches within walking distance of town have bathrooms? If not any beaches on island with bathrooms? Thank you.
Surfside, Jetties are first beaches that come to mind.
You might find this beach guide helpful. Click on North Shore Beaches to read about Children's (a smaller beach) and Jetties beaches. Both have bathroom facilities. https://nantucket.net/beaches/map.php
As a previous poster helpfully wrote, you can catch the Wave bus to Jetties and other island beaches. You could walk from town but it can be difficult if you're hauling gear.