Could be either or both.
Could be battery dead not holding enough charge to start it or could be the alternator not putting out enough voltage to charge the battery enough to be able to start it.
When the van is running, put a circuit tester across the battery and see how much the alternator is charging. It should be above 13.8 volts, ideally between 14.2 and 14.4 volts with lights etc on. (Those figures are about right top of my head).
If it's charging ok then, time for a new battery. A top up of di-ionised water may temperarily help.
If you test the battery on its own a good battery should hold 12.6 volts. I think anything below 12.2 is knackered.
Hope that helps, Ben.