The common dirt I have successfully removed by stuffing the carpets into
the washing machine.
This works for the carpets on the front wheel wells, the carpet on and
between seat pedestals, and also for headliners.
I even pulled the velour off the seats and successfully washed them
clean (incl rear bench!!!).

I am struggling with the main front carpet, because my machine only has
a 5kg drum. It´s not the weight, but the size you can fold the carpet.
I had my mom wash the front carpet in her 7kg machine. She refused when
asking for a second carpet though :-(

The backside of wheel well etc carpet with the black glue (I think it´s
Sikaflex) stays intact, at least if you cover those areas with duct tape
before washing. So you can just glue the carpet back into the Vanagon.

Regards
Raimund