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I always used the bus, but in your case, probably not practical; try this taxi company () - there are others, Google is your friend.

Uber and similar driver-operated ride share services are illegal in Germany, because incompatible with German labour laws.

The company Uber operates a limousine service in some cities, which is similar in scope and costs to other such sevices and only slightly less than a taxi. And taking such indvidualized transport options in Germany is always a lot more expensive than you imagine - that's why everyone takes buses and trains!

If you need a van, I recommend you book ahead, as it is unlikely that vans are available on demand at the airport!

The Berlin Airpeort BER like most in Germany is well connected to the city by the S-Bahn. One cannot take large furniture but luggage and bikes are allowed in the S-Bahn and U-Bahn, but buses can be problematic.

The potential problem is getting all of the things in and out if the kids are too small to help. But one can simply ask people around for assistance. I myself usually volunteer help when I see such a situation. The average German might not but would usually help if asked, especially when kids are involved. Germans tend to have empathy but not get involved in other people's business unless requested. Ideal would be if you know anyone in Berlin who could meet you at the airport to give a hand.

I personally think this might be better than the trouble of renting a van. One would have to transport all of their things to the car rental counter. Then drive through a city they don't really know, find parking (much easier said than done), unload and then drive the vehicle back to a rental drop-off place.