Allow users to specify a list of domains, or ideally url patterns, to limit search
As I understand it, your URL has a limit of 2048 characters. Obviously when searching a small handful of sites or the entire web that is more than sufficient. However, if a user wants to curate a large but selective set of sources, the space will quickly be filled up with site:one.example.com OR site:two.example.com OR etc..
One way to get around this is what Google does with Custom Search, by allowing the user to specify a url to a file, which is expected to contain some XML including a set of URL patterns. The search results are then limited to those URLs. I'm sure there are other ways. Alternatively, although less elegantly, you could just allow much longer input urls.
To customize your search better, there is also Bing Custom Search:
If you are using the GET method, you are limited to a maximum of 2,048 characters, minus the number of characters in the actual path. However, the POST method is not limited by the size of the URL for submitting name/value pairs. These pairs are transferred in the header and not in the URL.
I have see suggestions to TinyURL or Snipurl to extend the length out to extend the length beyond this limit.