You can enroll in their affiliate program and then use a RESTful API. Useful info can be found here. Basically, you apply and, if approved, you get all needed credentials and can find the key in their Partner Portal.
Then you can enter a testing environment to try you requests and configure the integration.
The final step is to ensure that your site meets Expedia requirements. You’ll undergo their review prior to going live and then you’ll start working with end-users.