Henning Makholm's answer addresses the issue of your possible statelessness, and suggests a way of obtaining a travel document.
The other alternative is to acquire a citizenship. Your case is complicated and I would recommend consulting an immigration lawyer. Below is some relevant information, mostly extracted from Wikipedia. Note that it is possible that you are not in fact stateless (e.g. if you are considered a citizen of Thailand or Vietnam), but merely lack documents to prove your citizenship.
Thailand's nationality law is based on the principles of both Jus sanguinis (citizenship through parents) and Jus soli (citizenship through birth). Your parents were not Thai citizens, so that leaves birth. This paragraph provides a summary.
Wikipedia doesn't have an entry for Vietnam's nationality law, but searching Google produced hits for me that suggest that the principle of Jus sanguinis (citizenship by parents) applies, so you may be considered a Vietnam citizen regardless of the fact that you were not born there.
There was a real missed opportunity here as you and your family would have been entitled to acquire citizenship after being resident here for 9 years (known as naturalization) but as you have permanently become non-resident since then, that probably doesn't count for anything anymore. On the other hand, you say you have a travel document jointly with your mother. What kind of travel document is it? Presumably you used this document to enter the USA. How did you address the issue of your citizenship at that time?
You didn't specify how long you have been resident in the USA for, but you should be able to naturalize as a citizen if you have been a permanent resident for 5 years. If this is the case, this is probably the easiest option as you can easily establish your case.
Dual / Multiple Citizenship
As it is possible that you may be eligible for two or more of the above citizenships, you may want to carefully consider your options. The laws regarding multiple citizenship can be complicated and vary for each country. Many countries, and particularly Asian countries, forbid multiple citizenship. So e.g. by taking up USA citizenship you may cut yourself off from the others. Whether or not this matters to you depends on what your long term plans are and where you intend to eventually settle. One downside of USA citizenship (and almost uniquely in the world) is the requirement to file US tax returns for the rest of your life (while you remain a citizen), even if you stop residing there (you may or may not have to actually pay US taxes, depending on your income and other circumstances).