What is the most effective way to fumigate fabric luggage?

Where do you live? Can you put it out, open, to sit in the sun for several days?

I don’t think fumigate is the word you want, unless you are exterminating insects. It sounds like you mean “deodorize.”

For deodorizing, it all depends on the source of your stink, and if how health oriented you are.

You could try misting it with a light dilution of ammonia and water. Ammonia is often used as a deodorizer. Make sure it’s not too strong.

Another idea is to dust the whole bag with baking soda, let sit, and vacuum it off (also a good deodorizer).

Or you could use Febreze.