Hi Matt,
The procedure is really simple:
1. Create a search form and set Form Method = GET in the Form Settings box;
2. In the front-end, use your search form to run a search;
3. Note the URL in the browser address bar, you can reuse that URL in a link to duplicate the same search.
You can now remove the search form display in the front-end, but don’t remove the search form itself on the admin side, otherwise your search link will no longer work.