What are some ways to season that don't rely on garlic and onions?
Aliums (the garlic and onion family) can be a trigger of both allergies and food intolerance.  Unfortunately, they're two of the most common flavorings in foods.  There's already a question on here about removing aliums: Replacement for alliums?

Tomatoes, are a separate problem, but it's not particularly prevalent in Asian cooking, other than on the Indian sub-continent.

I'd recommend looking for Jain cooking (an Indian caste that doesn't eat anything that's grown below ground), as there are lots that don't use tomatoes.

You might also want to look for FODMAP-friendly recipes.  It's a new-ish restrictive diet for people with severe food intolerance.  It's not 100% alium free, though, as they allow garlic flavored oil (so long as you remove the garlic) and they allow the greens from the alium family.  After you've been off aliums for some time and your wife isn't having any problems, you can try adding back one or the other and see if your wife can tolerate them.

As for your comment about vinegar -- sour is one of the under-appreciated flavors, and it can really help to perk up bland dishes.  You might want to consider citrus juices, such as lemon or lime to add more flavor to dishes.  Pineapple and other tropical fruits may also help.  But I'd also be careful about too much acidity -- that might be why your wife has issues with tomatoes.  (And if it is ... you might try finding yellow tomatoes at farmers markets, as they tend to be lower acid.)