Unfortunately, you cannot book transport including driver via billiger-mietwagen.de.
You can find vans with drivers at courier services or via various real estate portals. Make sure you use professional providers to avoid cost traps and poor service.
In general, the same criteria apply to the rental of a van with driver as to that of a rental van. Estimate in advance the exact duration of the deployment or the kilometres to be driven and make sure that the load is sufficiently fastened so that nothing happens to the car or grandma's heirlooms. With some providers, you can book additional removal materials such as tensioning straps or covers – even if you only rent the van.