That is a great question! Judging by your screenshot, it looks like that Search link is indeed in the footer menu of your store. You can add/remove entries in this menu (and also your header main menu) from within the Online Store > Navigation section of your Shopify Admin.
Once you are in that section, you can click into the menu you want to edit; which in this case is the Footer. From within the Footer section, you can feel free to edit or delete each menu item. You can delete the search entry here if you wish like in my screenshots below. For reference, we have a handy help doc which explains how Menus & Links work on your store - feel free to use this as a guide also!
That should do the trick to get that search link removed!