Allow iframes from specific sites?

I’d register an embed handler with wp_embed_register_handler. This gives you the added benefit of being able to just copy and paste the url into the editor as well as seeing a preview of the iframe.

add_action( 'init', 'se238330_register_embed_handler' );

function se238330_register_embed_handler() {
    wp_embed_register_handler( 
        'joetek',
        '#http://subdomain.yourdomain\.com/(.+)/?#i',
        'wp_embed_handler_joetek'
    );
}

function wp_embed_handler_embed_name( $matches, $attr, $url, $rawattr ) {
    $embed = sprintf(
        '<iframe class="joetek-website" src="http://subdomain.yourdomain.com/%1$s/" width="600" height="400" frameborder="0" scrolling="no"></iframe>',
        esc_attr( $matches[1] )
    );

    return apply_filters( 'embed_joetek', $embed, $matches, $attr, $url, $rawattr );
}

The code above assumes that joetek is the name of your embed 🙂 You’d need to update the regex in the second parameter of wp_embed_register_handler as well as in the callback function to match the pages you want to be able to embed on your website.