Early morning airport transportation with young kids

We have an early morning flight out of Reykjavik and I'm wondering how other families have handled getting to the airport in the wee hours of the morning. Our flat is pretty close by, but with two young kids and luggage, I'm not sure we'll be up for the walk. It doesn't look like the busses run that early and I'm not sure how taxis would want to handle taking two Littles. We will not have our own car seats.

Can anyone recommend a car service we can book in advance? Or have advice on how you handled it?

Thanks so much!