Hi NasH,
According to your document, ASP.Net developer created some API. You have to consume those API in your project means from php code. The API may return XML data. In your document there is API authentication, url, param, etc. You have to use those information to call the API from php. Check below link it may help you.
Consuming ASP.NET Web API services from PHP script
For more details, please search how to consume a API from php.