I've always found Universal to be on par, if not better than Disney in regard to food allergies, and we don't contact them before we arrive. I'm not saying it is a bad idea in the case of multiple allergies, but just a peanut/treenut allergy they can handle with no notice, in my experience. Many TS places have allergens pre-printed on the menu, or give you an allergy menu. Mythos comes to mind.
It is also possible to find some prepackaged snacks all over property, so reading labels is sometime an option. The Aventura food court comes to mind, also the QS place in Sapphire Falls. Aventura sells an array of milk alternatives, for example.
Another option we find to be super is staying club level at any of the deluxe Universal hotels. Like WDW, you just discuss your needs upon arrival, and they'll make sure to have something available for you.
Fourth, there is a Whole Foods located JUST off Universal property. (Sand Lake Road) We usually have a car, but the towncar services allow a grocery stop, or it would be a short taxi/uber ride from any US hotel. The Whole Foods has just about every allergy-free item that exists. It also has an extensive array of prepared foods, hot bar, fresh juice bar, barista, etc. Not inexpensive, but you can easily get many allergen-free foods there. If you are coming from WDW or the airport, it is very easy to get there. Just make sure your hotel room has a fridge!