The obvious way is to have them planning / committing a robbery during the Chitauri invasion, and either fighting the aliens or finding a way to carry on with the crime with total chaos erupting around them. But that's either a cliche or looting, and neither much appeals.
Another possibility is that they hear about a mansion loaded with antiques and decide to pay a visit - unfortunately they don't know that Tony Stark has remembered the old family home, is about to use it as Avengers HQ and has already installed JARVIS. But that probably ends up as a Home Alone situation and eventually with them in jail, not the most exciting plot. Lots of variants on this, of course - the one I think might be interesting has them stealing Dummy and ending up having to pay Tony to take him off their hands, a la Red Chief.
The really silly one isn't an Avengers plot - I just remembered that Dortmunder's girlfriend is named May. So what if she was Peter Parker's Aunt May, and she and Dortmunder ended up bringing up a really bright kid who would one day have superpowers? Does Peter follow in "Uncle John's" footsteps and become a moderately competent supervillain, or does he still become a crimefighter and end up webbing Dortmunder?
There are other possibilities - I'm very tempted to write this as a "five things" story or something, but already have too much on my plate. So if anyone wants to take one of these ideas and run with it please be my guest, but an acknowledgement would be nice.