I have the same problem as described quiet often here on the community pages already. We have a multisite setup (with subdirectories), run on the newest WP and Marketpress Versions.
I first had the problem all over the store pages, that once you hit checkout all items in the shopping cart disappeared. After including the COOKIEPATH into the wp-config.php (as described here: https://premium.wpmudev.org/forums/topic/help-there-are-no-items-in-your-shopping-cart) it is working fine in the main store, but not in the multisite sub-stores.
Any help appreciated! BTW, I already deactivated all plugins! And the .htaccess is fine, too!