“It’s time for … “ usually has a different meaning. It’s time for dinner/bed/class to start/. Using it in your sentence where you intend a different meaning is unnatural. You can say either ‘a time’ or ‘the time’ depending on how much you want to emphasize the importance of summer in this regard.

Both are correct, but they have slight differences in nuance: 1. "It's time for them to have fun." - Suggests that summer vacation is the specific period when children can have fun. 2. "It's a time for them to have fun." - Suggests that summer vacation is one of the periods when children can have fun, implying there could be other times too. Both are natural and depend on the context you want to convey!

"A time" is better because it specifies that "time" refers to a specific time: "summertime". Without the article it is still correct, but the two sentences are not tied together as well.

Both are OK; the first [without the article] is more 'spontaneous', as though the children themselves were saying it. The second [with the article] is more formal. In both cases you can omit 'for them' : 'It's (a) time to have fun'.