Petitions are no substitute for market research.
Vegans are not going to like In-and-Out Burger. They will not eat there, even if there is a vegan option.
These ideas fail for two reasons:
1.) People who avoid meat for ethical reasons won't eat a vegetarian burger that was cooked on the same appliances as a meat burger.
2.) People who avoid meat for health reasons don't eat at fast-food burger places.