I'm 'just' gluten free but in my experience Disney is second to none in accommodating those of us who have dietary restrictions. That being said multiple allergies can be more difficult, whether it's one person with multiple allergies or a group of people with one allergy each.
Here is what is on the DL web page for multiple allergies followed by the link to their dietary needs page. I'd suggest you use the email they provide to discuss your situation:
"Guests with requests associated with the list below should contact us at Special.Diets@disneyland.com after booking all dining arrangements for their visit at least 14 days prior to arrival.
- Any allergy or intolerance not listed above
- Individual Guests with 4 or more allergies/intolerances or several Guests with allergies/intolerances within the same party
- Metabolic disorders
- Phenylketonuria (PKU)
Our team will assist Guests in navigating the dining process. In addition, Disneyland Resort restaurants will do their best to accommodate dietary requirements but cannot guarantee that they can meet all requests."
https://disneyland.disney.go.com/guest-services/special-dietary-requests/
You don't need to book all reservations ahead before you contact them. They can help you work out which eateries will be able to accommodate your various needs. But be aware that
as a rule they won't go item by item over the menus to help you choose what to eat. They'll just help you decide
where to eat. What to eat will be something you'll work out once you arrive at the eatery. Once you've decided where to eat, look at the menus online and see if there's something that appears to 'fit' or would 'fit' with a few adjustments. Personally, I try to find two things that 'look do-able' ahead of time. Some of the allergy menus can be found online, but you have to google them (i.e. 'Blue Bayou allergy menu' or 'Disneyland allergy menu') as they are on third-party websites and blogs. They are NOT on Disney's official site. At least they weren't last time I looked. But be aware that items/suppliers change so be prepared to be flexible.
Table service usually is your best (albeit most expensive) option. Chefs there have a range of ingredients and can usually whip up something 'off menu' if there's nothing suitable on the menu. CS and QS are pretty much stuck with adapting the items on the menu.
And yes, do bring some 'safe' food. We can't just grab something 'willy nilly' from a street cart to 'tide us over' in the middle of the day. I admit to getting a bit 'hangry' (or more truthfully 'hupset') when I let myself go too long and then couldn't find something 'right then'. Now I always carry a couple of GF snacks, just in case.