"What do people enjoy doing for the vacation seasons recently?" and "Recently, overseas travel during the vacations is popular" are both examples of terrible English.
"For the vacation seasons" is very unnatural and, in BrE, "during the vacations" is too. Both examples use the wrong tense.
What have people started to enjoy doing for their vacations recently?
What have people been enjoying doing on holiday/on vacation recently?
Recently, overseas travel for a holiday/vacation has become popular.
If that's the general standard of the English in your ESL book, I suggest you get a new book.
Student or Learner