Hi there.
The 10.x licenses will not work with 100.x.
Whether you need to buy another license depends on what capabilities you are using in the Runtime. See Licensing capabilities for more details.
In short, if you're not doing any on-device visual analytics (like GPU viewshed), not editing data (unless it's public), not using the Navigation API, and not using local files like Shapefiles or GeoPackages, chances are that you will be good with the Lite level license. But do check the table I linked to above.
You can find your Lite level license key at the ArcGIS for Developers dashboard. The Lite license key is free to use for an unlimited number of deployments.
If you need a higher level license than Lite, then you will need to buy new licenses, or use ArcGIS named user login to enable capabilities according to the user type. You can see that outlined in the licensing table too.
Hope this helps.