If the problem is the computer like I think it is, the reason it runs eventually is because it has warmed up enough to work correctly. Even leaving just the key on for a bit will warm it eventually due to the electricity running through it. Although as winter approaches and the weather gets cooler, the problem will get worse since it will take longer to warm up the computer.
Tomorrow morning try the hair dryer trick and see if the van starts right up after you warm the computer. If it does, you will need to replace it. Depending on where you buy it, you can get a used one for $100-$150, or a rebuilt one for around $300, instead of dishing out $600 for a new one.