I don't know of any site that uses % in the url >.<
% signs in a URL are used for escaping certain characters. In the given URL, for example, %2F is character code 47 which is the forward slash ("/"). naturally that cannot appear as is since it would be another slash and "break" the url.
This is also part of the problem in the original post. the date is being represented as 6/13/2012, but with the slashes escaped. I guess the problem here is that the % signs have special meaning on the windows command line.
You can escape them by inserting a caret before the percent sign.
https://site.gov/blah/blah/decision?process_date=6^%2F13^%2F2012^&v_dispatchid=#######^&but=Reports
(ampersands have special meaning too, so those need to be escaped).
For getting the proper URL for the previous date, you can probably do some string processing on the output from date /t; though it would likely rely on the date format being used.