Yes and No.
Currently, WooCommerce API does not support a way for us to post the tracking number on to an order. WooCommerce API only supports updating the status of the order. This means that there is no way for Ordoro to directly post a tracking number on a WooCommerce order (Note: Ordoro does post tracking numbers for orders coming from other channels like Shopify, BigCommerce, Magento, eBay, Amazon etc). We have partnered with a company that has built a plugin for this feature, and you can find more details for that here.
Currently, Ordoro will add a note to the order in WooCommerce with the tracking number information inside that note. Ordoro will mark that order as completed. Here is what the note looks like. With the latest version of WooCommerce (2.2), WooCommerce will email the customer that note.
If the order was already completed in WooCommerce before Ordoro could post that note, then that note will be ignored by WooCommerce.