When it comes to Ajax and promises, React recommends the usage of the new fetch API. Although it's browser support is still not widespread, you could make up for it by using a polyfill as @mayank mentioned.
For our projects, we use a mix of fetch and jQuery's $.get/$.post. The new components use fetch while the existing old ones use jQuery's Ajax API.
I would recommend you to steer clear from other Ajax libraries for new code and use the fetch API as it is very lightweight, has simple syntax which is easy to read, and will be natively available in all major browsers in future.