First rule of troubleshooting, check the easy things first.
Easiest thing to check that would cause the symptoms you describe— water gushing at first, then stopping— would be a clogged inlet screen. Right where the hose attaches to the trailer, there is probably a rubber grommet to make a watertight seal, with a wire mesh screen either attached to it or right behind it. Unscrew the hose, and see if there's a screen, and if there is, if it's clogged.
Many Airstreams have a built-in pressure regulator close by the inlet, to protect the interior plumbing from high water pressures. It's easy for that to clog, which is one reason why there should be an inlet screen to keep debris out of the pressure regulator. So, if there's no inlet screen or if it's damaged, the pressure regulator might be clogged.
While you've got the hose disconnected from the trailer to check the inlet screen, also make sure you're getting water through the hose. The problem might not have been in the trailer at all, it could be at the other end.
Once you've cleared those two hurdles, then it's time to start looking for valves. Bypass valves and shut-off valves should be open, low-point drain valves should be closed.
Does the problem happen equally with all faucets? Does the toilet flush? If one faucet works and another doesn't, then the problem can be found fastest by starting at the non-working faucet and working backwards. If the problem affects all faucets and the toilet equally, it can be found fastest by starting at the water inlet and working forward because the problem occurs somewhere before all of the branch lines split off to the various fixtures.
Does the problem happen with both hot and cold water? If hot water works and cold water doesn't, or vice versa, then you start at the faucet (again) and follow the line that doesn't work until you find the problem.
Once you've found the source of the problem, fixing the problem is another matter best left for another post.
Engineering: Finding complex solutions to simple problems you didn't even know you had.