functions.php

<?php

/**
 * Template Parts for Display Posts Shortcode
 * @author Bill Erickson
 * @see https://github.com/billerickson/display-posts-shortcode/wiki#using-template-parts
 *
 * @param string $output, current output of post
 * @param array $original_atts, original attributes passed to shortcode 
 * @return string $output
 */
function be_dps_template_part( $output, $original_atts ) {

	ob_start();
	get_template_part( 'partials/archive', get_post_type() );
	$new_output = ob_get_clean();

	if( !empty( $new_output ) )
		$output = $new_output;

	return $output;
}
add_action( 'display_posts_shortcode_output', 'be_dps_template_part', 10, 2 );

Where would I put this in the plugin?

Anywhere. I think functions.php references the theme. Add this anywhere to the functions.php