Has nothing to do with SolidWorks Premium. Anyone wanting to use SimulationXpress has to go through this. If you have Premium, just use the full Simulation tool and skip SimulationXpress altogether
So it's time for you guys to upgrade to premium - Never tried and not sure how much you can do with express version
Not really upgrading to premium is something that a company decision based on their needs for example if we have a 4 seat of solidworks standard and 1 seat of premium the 4 seat can model part and used the express to visualize how the model will behave as initial checking while the premium will use for much in depth analysis, that way it could save some money to the company because in reality not all the 5 seat will perform analysis that needs a premium version at the same time
When you start SimulationXpress, a window will open requiring the Product Code to be entered. In the same window, click on the link 'My.SolidWorks.com/xpress'. This will open a new window and provide the 'Activation Code' which can be copy/pasted into the Product Code field to enable the Xpress product to run.
Your other option is manually get it:
1) Go to https://my.solidworks.com/xpress in a web browser
2) Log in using your credentials if prompted
3) Enter your SolidWorks Serial Number, the version, and the type of Xpress product to get the code.