I'd like to provide a URL such as "http://blahblah.com?search=%s" where Firefox replaces the %s with the content of the search box.
You can do this with Firefox! Add a bookmark with that URL, where
%s is the search query, then simply set a keyword for the bookmark. You can then search using
<key> <search term> in the address bar. For example, the bookmark
http://www.google.com/search?q=%s with keyword
g means you can type
g stack overflow in the address bar to search Google for "stack overflow".
Alternatively, go to the site you want to search, right click in the search box and click "Add a keyword for this search".