Solution
The first step would be for you to go through the troubleshooting of checking if the assigned license matches the one that you are launching from the client's side. You can refer to the link
Volume License Manager (VLM) Troubleshooting Guide: Main Page to ensure that you touch all the basic points.
If everything is good, you might be having incompatibility issues between the NI License Manager (NILM) and VLM. One of the symptoms is that you can properly connect to the VLM, and see that the NILM shows a green indicator to the license, but you cannot launch the software.
The compatibility of the NILM and VLM can be summarized as below:

Once you make sure the NILM and VLM are compatible, the license should be pulled out properly from the server and the software can be launched again.
The recommended way is for both NILM and VLM to update to the latest version which will be future-proof. unless you are limited with Windows capability.