Regardless if you are shipping with your account, on behalf or billing a third party, the tracking numbers should be updated in the WMS either when processing the order (as long as the WMS supports it), or after you manifest (supported by all WMS integrations).
If you see the tracking information in techShip but not in your WMS after manifesting, this could relate to several issues, ranging from system issues to human error.
To make sure everything has been done properly prior to reaching support, verify that:
1) You processed and not voided the orders
2) The orders have been packaged in the WMS (3PL Central only)*
3) The orders have not been closed before manifesting in the WMS (all WMSs), since techShip cannot update any information to closed orders
4) Manifested (Closed) the orders as described in this article: 02 - How to manifest [Step-by-Step guide]
5) Re-ran WMS Update in case there was a connectivity issue, as described in How to Re-Run techSHIP WMS Update processes [Step-by-Step Guide]
6) Generated a Shipment Detail report (as described here: How to generate and read Shipment Detail reports [Step-by-Step guide]), and confirm that the WMS Update Status column (typically column AM) shows Completed instead of Pending