My understanding of the cap economy is that it was established and maintained by the Hub in Fallout 1. Given that, canonically, how did that particular barter system become the de facto system on the east coast, specifically in the Capital Wasteland and surrounding environs (The Pitt, Point Lookout, etc.)?
Presumably the same way the Brotherhood of Steel made it out East. People moved, and brought the use of Caps as currency with them. Keep in mind, Fallout 3 occurs some 200 years after the bombs fell, and roughly a century after the events of the original Fallout.
According to the Fallout Wiki, no explanation is offered as to why Caps are used in FO3, but presumably, given that the NCR has switched to it's own currency in FO2, this reflects how backward the DC wasteland is relative to the somewhat more rebuilt territories out west.
According to the Vault
There is no explanation whatsoever given for the use of bottlecaps in DC.
3But at least caps were in the area -- hence, the Nuka Cola Plant Aug 30, 2010 at 1:16