Integrate with the API

Hello.
We need to do integration with the yclients API of our directory site: https://top-barbers.ru/
Site on CMS Wordpress Advanced Custom Fields PRO.

An example of implementing the same task on a competitors website: https://top-barbershop.com/place/oldboy-sevastopol...
That is, we want the pages of barbershops on our website (for example, such https://top-barbers.ru/barbershop/barbarossa-na-yasenevo/) to be able to sign up for a service through yclients. I attached a screenshot of what the final result should look like. Please note that we have a catalog site and we aggregate a lot of objects, so this script should work for all barbershops.

Plus, we need minimal reporting to track leads: which barbershop received the application, phone number and customer name

We have a developer account with YCLIENTS and access to the API.