There are no associated settings or permissions for that search box. It should always be there.
It's probably a bad template, bad file, or some customization.
1) Visit /install and click the button to rebuild the master data. That will restore the original master templates in the Admin CP which may fix the problem.
2) Run a file check and upload the original files as necessary to fix any errors:
Admin CP -> Tools -> File Health Check
3) Try disabling your addons from the home page of the Admin CP to see if that fixes the problem.