No, Shake Shack does not currently have the vegan alternative to meat ‘beyond burger’ on their menus right now but they are working on expanding their food menus to be more inclusive for vegetarians and vegans.
Shake Shack does already have vegetarian burgers on the menu called ‘Veggie Shack’ and ‘Shroom Burger’ but they’re not vegan as they also have ingredients like mayonnaise and cheese in the bun. Some Shake Shack stores serve vegan mayo to go with the Veggie Shack burger so you’ll need to check in the store to find out which one they’ll use. You can remove the cheese and also ask for a gluten-free bun (the original bun contains dairy) to make the Veggie Stack burger suitable for vegans. The Shroom Burger cannot be made vegan though as the portobello mushroom used is stuffed with cheese.