There have been other posts on this forum with suggestions for sightseeing in Lyon. Re vegetarian and gluten free: I can't advise on gluten-free, because it's not something I personally am concerned about, but my guess is that it might be tricky. There are gluten free products in supermarkets these days but I don't often see it marked on restaurant menus. Re vegetarian: I'm more or less vegan, and just spent three months in Lyon, so here are a few ideas. None are strictly vegetarian, but they're places I've gotten good meals.
- Le Jardin des Délices, 17 boulevard Yves Farge 7ème arrondissement: wine bar and restaurant, almost always offers a "plat végétarien"
- Les 3 Chauves, 88 rue Bechevelin, 7ème : casual restaurant/bar, excellent brunch with veg options
There's a group of "Brasseries" created by Paul Bocuse that we like for moderately-priced versions of fairly traditional French cuisine. The one that most reliably has good veg options is Le Sud, 11 place Antonin Poncet, 2ème. (very centrally located). L'Est, 14 place Jules Ferry, 6ème, also offers veg choices. Another advantage (for me) of these brasseries is that they start dinner service at 7:30 pm, whereas some restaurants in Lyon won't serve you until 9 pm.
Pizza is a good option for vegetarian but gluten-free crusts are rare. I have seen them offered but can't remember where.
Good luck! Lyon is a great place to eat even if you have constraints.