The only check valve inside the trailer plumbing that I'm aware of is at the hot water heater to keep the hot water from pressurizing and pushing back into the cold input line. If you are not the first owner then it's possible someone installed a backflow valve to the inlet, but usually the manufacturer does not.
If there is any type of in line filtering then there may be something there blocking the flow. The screens on the end of your faucets can get clogged with sediment but that would usually only effect one faucet at a time and not the whole trailer.
I would still try filling the freshwater tank and see how that works with the water pump. It may rule out a few other things.
Keep us posted as to what you find.
