Assuming that those users have accounts on your site (are subscribers, authors, or some other role), you can add custom fields to their profiles. The easiest way is to use a plugin, like Advanced Custom Fields (https://wordpress.org/plugins/advanced-custom-fields/).
The fields would be available in their profile pages, and you can add, and use in any template.
I would give you code examples, but their site seems to be down… I’ll try to update the answer once it’s up again.
Update: ACF could take care of creating the fields, and displaying them in the user profile page, and you can use add_user_meta to save them:
add_user_meta( $user_id, 'custom_field_name', $search);
If you don’t want to use ACF, you can display the fields in the profile with custom code. Deleting the searches would be a bit harder, though, but doable.