On the location bar you need to use a keyword search if you want to add such parameters.

You can use a (one letter) keyword search to ensure that Firefox does a search and doesn't threat the word as an URL.
You can either do that for existing search engines via the Manage Search Engines window or create a bookmark with a keyword assigned to it.

• http://kb.mozillazine.org/Using_keyword_searches

Thanks for the solution, but that just seems unnecessary. If other browsers can know the difference between and URL and a search string with parameters, then there's no reason Firefox shouldn't be able to as well.

It doesn't seem like it would be all that difficult to implement.

The awesome bar has its own features to try to find suitable links and is already customisable.

• see Address bar autocomplete suggestions in Firefox
• and Search the web from the Address Bar

It doesn't seem like it would be all that difficult to implement. 

Yes that is correct, because it is already implemented, just key your example "site:support.mozilla.org browser" in to the search bar !
Note again that is flexible and customisable

• Use popular search suggestions in the Firefox Search bar
• Search bar - add, change and manage search engines on Firefox

I don't want to have to change focus from the address bar to the search bar. That is the whole point of the convenience of having the same field handling both search strings and URLs. You open up a browser window or new tab and just start typing without any other input. If I'm already going to be manually clicking into a search field, I might as well click the one in the numerous start page addons that could accomplish the same thing.

I saw where you can designate a character to signify a keyword string so that the address bar ("Awesome Bar") doesn't error out, thinking the keyword + parameters was a URL that didn't work. But again, it just seems like an unnecessary step that other browsers don't make you take while accomplishing the same goal.

I want to open a new window or tab and start typing the search string or URL, as is possible in other browsers. I don't see the need to complicate things.

It should work if you first type the search terms and put additional parameters like site:xxx.com at the end.

• browser site:support.mozilla.org

You're right, that does work. I've always started with the parameters when I search. I guess that's an easy workaround for now until or if they ever allow you to start with parameters instead.