Can anyone help with advice on how to integrate API for my client?

I have a client looking to build a website selling products from a subscription and authorised account-holder products distributor. The company she sells products on behalf is called Alumier; they sell prescription-based products that require the user to have undertaken a consultation first, prior to being provided 'access' via a portal. The company Alumier have their own API key for integration into their website, however I have no experience in building this sort of website, or using API keys. I've uploaded the documentation that I've been provided by Alumier below, which should give you enough information. I'd appreciate any help, or advice you can give. Thanks.