They're all grammatical, and differ slightly in meaning:
I think they take their son to school by car
In the present, I believe that on a regular basis they use the car to take their son to school.
I thought they took their son to school by car
In the past (probably but not necessarily recent, for example in reply to someone saying "their son took the bus to school this morning"), I believed that either in the recent past, on several occasions, on a regular basis or on one specific occasion being discussed, they used the car to take their son to school, but don't necessarily do so any more.
I think they took their son to school by car
In the present (probably as a reply to a question just asked like "how did their son get to school?"), I believe that they very recently used the car to take their son to school.
I thought they take their son to school by car
In the past, I believed that on a regular basis they use the car to take their son to school, and continue to do so now. You'd probably only use this construction in reply to someone telling you that their son regularly gets to school via a different method.