Kayak has an affiliate program that you must enroll in before integrating their API. Keep in mind that Kayak doesn’t permit integration unless your platform has more than 100,000 monthly visitors.
If you have more, you can use their API or white label. To proceed you have to define which kinds of search data you want to receive and contact them directly.
If you have fewer than 100,000 monthly visitors, Kayak offers an affiliate programs trial using third party affiliate networks like CJ or Webgains. They will connect you with smaller brands belonging to Booking Holdings, like Momondo.