With the help of a third party app you can allow searching for listings on your Facebook page!
Static HTML https://apps.facebook.com/static_html_plus/?fb_source=search or,
Static iFrame https://apps.facebook.com/iframehost/ will allow you to add custom code.
All you need to do is copy the iFrame HTML code from the IDX Manager and paste it into the app.
See it in action here: https://www.facebook.com/idxsolutions
Depending on which option you chose above you will paste your iframe in one of the following:
Clicking on Tab Settings will allow to change the tab name from Welcome to something of your choice.
Clicking on Actions will allow to change the tab name from Welcome to something of your choice.