I will integrate rest api into your website using php

Its a set of functions and procedures that allow the creation of applications which access the features or data of an operating system, application, or other service.

REST stands for Representational State Transfer. It relies on a stateless, client-server, cacheable communications protocol -- and in virtually all cases, the HTTP protocol is used. REST is an architecture style for designing networked applications.

SOAP (Simple Object Access Protocol) is a messaging protocol that allows programs that run on disparate operating systems (such as Windows and Linux) to communicate using Hypertext Transfer Protocol (HTTP) and its Extensible Markup Language (XML).

I can give you suitable time and budget estimate. I'll have take a look of API documentation before giving time estimation.