Inventory is kept in sync between the retailer's and the supplier's store. The supplier controls the inventory levels and ships the products when something sells through the retailer's site. The retailer does not handle, purchase, or control any inventory as part of this partnership.
If you notice the supplier's items are out of stock, please reach out to the supplier directly for an update on the inventory levels. You can find the best contact information for your partners, in your "My Partners" tab.