There are quite a few budget ultra short throw projectors out there, just none in the 5000 lumen class. You can use projectorcentral.com's 'search by feature' to search for 'short throw'.
Probably the closest fit for your application is the BenQ MP776ST. 3500 lumen, 48" wide image from 2' 5" with a street price about half your maximum budget. It has vertical keystone, so you could 'try' a vertical stack for slightly higher than your budget, and get a theoretical 7000 lumens. I would hazard a guess that it would be a pain to converge a perfect image with such short throw lenses, but, if you're just showing a webcam, and lumens are more important, it may be worth a shot.http://www.projectorcentral.com/BenQ...ulator-pro.htm
To do this the 'right' way, you'd have to venture up to professional / staging class projectors from the likes of DP / Christie / Barco with prices several orders of magnitude above your budget.