We do it. We only have one child. We both work full time, the kid is in elementary school full time (kid is 4), and one of us bikes for transportation. The biker transports the kid daily, so it doesn't limit activities except that we don't sign up for things that are too far away (say, 3 miles, depending on the route) outside of the time where the driver could take the kid.
Really, in order to make this work, you're probably going to have to sacrifice something - time, energy, activities, comfort. Since we've almost always had only one car, we're used to scheduling around it and being slightly inconvenienced. It's just how we do things, and it is completely unremarkable once you get used to it.
How old are the kids? Are they getting to an age where they can be in charge of their own transportation (walk, public transport, bike, scooter)?