The FAQ on the website says what kinds of accommodations they can make:
Do you offer a vegetarian option?
Vegetarian menus will only be offered when they make sense with a given menu. Currently a vegetarian option for Paris 1906 – Escoffier at the Ritz is NOT available for obvious reasons. Future menus will include vegetarian alternatives and may be entirely vegetarian, so vegetarians will have the opportunity to dine with us.
What about allergies?
Known and limited food allergies will be accommodated as specified. Personal preferences and dislikes will not. For Paris 1906 - Escoffier at the Ritz we can accommodate the following allergies when noted at the time of ticket purchase… and only if purchased directly from Next: shellfish, nuts, raw alcohol, raw proteins, legumes, and pork.