How can a non-US crypto exchange legally onboard US customers?

You can do this three ways:

1. Obtain your own Money Transmitter Licenses, including the coveted BitLicense (for the State of New York)
2. Obtain your own Banking Trust Charter (NY will still not be covered)
3. Find a Bank/Trust/MSB who will be a ‘sponsor’ for your licensing requirements and they end up being the Program Manager, i.e. they run it under their license. Again, this would be limited to 49 US states only (NY is not included).

In order of priority, most people who wish to enter the US market #3 is the easiest and #1 the most difficult. Lease, versus Buy, versus Build. Try working with someone who can offer the service to you.

Example offering (full disclosure, this is my company): Bitcoin Money Transmitter License | Cryptocurrency License - Faisal Khan