You can certainly get a lawyer and sue the insurer. Or you can go to small claims court, if the amount is within the limits they set. I don't quite see the logic of denying the claim because of the lack of repairs. However, an insurance company might legitimately think it's "fishy" that you switched to full coverage and the van disappeared as soon as you did. "Fishy" in and of itself isn't a good reason to deny a claim, however they are certainly entitled to ask a lot of questions!

Think of ways you can help show that the car wasn't already gone when you bought the increased coverage. For instance, can your agent verify that he saw it? Are you a "frequent claimer" or is this the only time you've made a claim. Can a neighbor give you a statement about seeing the vehicle? One possibility is to offer to provide a sworn statement from you supporting your claim, called an "Examination Under Oath." (Before you do that, you should talk to a lawyer; if you are accused of lying under oath things could get serious.)

Not all insurance companies play fair on claims like this, and the shadier ones will grab any excuse. If you bought your insurance from a low-rated company that habitually denies claims (check the internet traffic to see if you see a lot of really ridiculous cases, bearing in mind that even honest insurers aggravate people sometimes), you should probably head straight for small claims court. Arguing with those folks can be a waste of time.

If you have a decent insurance company, I would suggest you collect up the best evidence you can find and ask your agent to go to bat for you. Whether he does or not, try to get the insurer to reconsider their denial. Ask to have the adjuster's supervisor review it, and insist that, if they don't change their minds, they give you a written reason for the denial, pointing to all evidence they are relying on. If nothing else, that will make it easier to sue them, since they can't belatedly think up excuses they didn't tell you about.

If push comes to shove, you have some kind of corroborative evidence that the van was still around on August 6, and you have the time and energy, try small claims court.