I'd like to get GPS all traces from this web service which is based on OpenStreetMap.
Is it possible? How to do that?
This service uses IMHO OSM just as basemap layer but the tracks itself don't rely on open data. Thus as @scai says you need to find an agreement with the pages operator.
If you analyse the website, you will see, that the tracklayer is embedded via an WMS. That means it comes prerendered to your browser and you can't extract the vector data easily.