Most think tanks and special interest groups track where reporters use sources from think tanks and special interest groups for quotes. Almost every one of these keeps track of where their name and members appear in the media. Go to the site for an interest group or think tank and look for pages on the site with a link title like "In the News".
Another way is to look for quotes in full-text sources. LexisNexis Academic is useful for this.
Academic Search Premier searches articles from magazines, journals and newspapers. It can be useful for finding articles from members of policy institute to see what positions they take on an issue and sometimes articles about the think tank. Try searching the name of the think tank in quotation marks.