How to get a JSON response from a Google Image Search?
You can indeed use the Google API for PHP. You just need to configure your custom search engine to search the entire web. To do this, login to your CSE settings at https://cse.google.com/cse/, and remove all sites. Then, change 'Sites to search' to 'Search the entire web but emphasize included sites'.Since you don't have any sites specified, it will just search the whole web.
You can test using the box on the right.
As for documentation, the CSE API is documented here https://developers.google.com/custom-search/json-api/v1/reference/cse/list#parameters