How can you disable search for non-members on your WordPress website.

Question

You can restrict any area with is_user_logged_in() function. For example: If you want to hide search for non-logged-in users then enclose search DIV inside this function like this

if(is_user_logged_in()) {
    <div class="search">
    </div>
}

Search for the DIV that contains Search Box and apply function as shown above. By this way you can restrict any area for non-logged-in users.

NOTE: Since you are new in PHP, Don’t forget to enclose this function inside php tags (<?php ?>). And also make sure you don’t open <?php for this function if it already being opened.

And about your Plugin suggestion, Plugin recommendation is off-topic here. We cannot suggest any plugin as per the site rules, So you have to find your desired plugin by yourself.