I agree with Nigel that the children would be happier on the train. If you book a car service, you will still have 2 transfers, from the van to the cablecar up to Grutschalp, and then to the little train into Muerren. (They usually transfer your luggage for you at that point). Or if you take the car service to Stechelberg, you board the Schilthorn cablecar and change cars at the Gimmelwald station.
If you want to use a car service, here are a couple I found:
https://www.alpinbus.com/transfer/zurich-airport-zrh/lauterbrunnen. 307 CHF quote
https://www.suntransfers.com/zurich-airport-to-lauterbrunnen This one offers car or van, 345-368 CHF
If you want to take the train, there are direct trains from Basel SBB to Interlaken, but you have to get from the Basel airport to the SBB train station. If you fly into Zurich, you can get a train right in the airport that will lake you to Bern, but you have to change trains there to reach Interlaken.
That change in Bern can be a beast at high season. We were heading to Muerren from Basel last July to meet up with my son and his family (two 6-year-olds, luggage, and a bike). They were coming from Zurich and the plan was for them to board the same train we were already on at Bern. The platform was so crowded it was chaos, and they barely made it on the train. But maybe it is not always like that.