In the USA, are there import taxes (or equivalent) between states? Or is it in effect a customs union?
No. Such interstate taxes are prohibited by the Commerce Clause as they would be "undue burdens on interstate commerce".
The Commerce Clause is an evolving doctrine that has been largely developed by the courts (in particular the US Supreme Court).
The most recent development was South Dakota v. Wayfair (2018). Section A gives a summary of this doctrine's development and states:
Modern precedents rest upon two primary principles that mark the boundaries of a State’s authority to regulate interstate commerce. First, state regulations may not discriminate against interstate commerce; and second, States may not impose undue burdens on interstate commerce.