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.)

Just leave them out - there's a lot of techniques both western and asian that just ignore them. In a sense the 'trick' would be to work out what you need and what you need to avoid from each of those and work around them.

Rather than relying on those 'common' base ingredients - it might end up being a good idea experimenting with different ideas.

As an indian - I'd automatically think of spices - There's a bunch of spices that work well and wouldn't add heat (if not needed), though you might need to work out what your wife feels comfortable with. Stuff like cumin and pepper come to mind, but you could do interesting stuff with common western spices like basil and thyme as well. Basically there's just a lot of options depending on what you're cooking.

In indian cooking we often temper spices or dry roast them with just a touch of oil in a pan. This kinda removes the rawness in spices (which in addition to making them smell extra good, might help with palatability here). You can also prepare more than you need and use what seems appropriate adjusting to taste.

You don't need a massive amount of spice to make something good. Let the ingredients you can use speak for themselves. .

If you need acid - there's other fruits and vegetables that can do this. Lemon would give a good hit of sourness which vinegar and tomatos would do. You could add the sort of bulk tomato (or onion) might give with say a mirepox style thing.

So basically take what you know - try to work out how to swap it out on the dish - maybe with more than one thing and see if your wife likes it .