This is how I create a promo code. I'm only using percentage off, but you can set it up however you want. I'm getting $_POST['percentage'] and $_POST['coupon-code'] from a front-end form, along with the nonces

create-shopp-promo.php

<?php
if ( isset( $_POST['action'] ) && 'add-coupon-code' == $_POST['action'] ) {
	if ( empty( $_POST['add_coupon_code_nonce'] ) || ! wp_verify_nonce( $_POST['add_coupon_code_nonce'], 'add_coupon_code' ) ) {
		fc_message( "You don't have permissions to add a coupon code.", 'error' );
	} else {
		$_POST['percentage'] = filter_percentage( $_POST['percentage'] );
		$_POST['coupon-code'] = sanitize_key( $_POST['coupon-code'] );

		$Promotion = new ShoppPromo();
		$data = array(
			'name'     => $_POST['coupon-code'],
			'status'   => 'enabled',
			'target'   => 'Cart',
			'starts'   => 1,
			'ends'     => 1,
			'search'   => 'all',
			'type'     => 'Percentage Off',
			'discount' => $_POST['percentage'] . '%',
			'rules'    => array(
				1 => array(
					'property' => 'Promo code',
					'logic'    => 'Is equal to',
					'value'    => $_POST['coupon-code'],
				),
			),
		);

		$Promotion->updates( $data );
		$promo_id = $Promotion->save();
	}
}