Well, actually, the best option would be to say,
They watch cartoons a lot.
"They watch cartoons for a long time" sounds a bit strange because "for a long time" implies some specific instance of something happening while Present Simple implies that something is happening regularly.
One could say,
They watch cartoons for long stretches of time.
"For a long time" would be correct in sentences like this,
— For a long time, this method was considered to be one of the best ones available.
— She has been ignoring me for a long time.
Note how in both cases we are speaking about a single instance of something—a single situation.
"A lot of time" would be correct in the following sentences,
— A lot of time has been wasted.
— I don't have a lot of time.
Note that "a lot of time" is not an adverbial with a temporal meaning in either of these cases. It is a subject or an object.
(And, "for a lot of time" is obviously wrong.)