In my experience, there are few factories still located in Taiwan. Most of the Taiwanese factories that I work with have moved all of their factories to China. The only factories I know that are still there are Ribbon factories.
You could contact manufacturing companies in Taiwan and Hong Kong and ask if they are looking for U.S. designers, freelance or permanent. Some of the factories I work with do hire Americans, mostly on a freelance basis. I would start with manufacturers in industries you have experience or interest in.
You might also want to contact a large import agency, such as Li & Fung or Testrite. I’m not sure who to contact there, but they work with thousands of factories in a variety of industries.
Working for a retailer/wholesale/trading company is the way that many designers gain experience working with factories.