this is possible. ajax is based on http so you can request with php just the same as your browser does on that ajax website. the 'only' thing you need to do is to analyze the request your browser sends to the server and pick out the right one.
this whole process tends to be network analysis and you will need a tool like a sniffer or a proxy for that.
for the php part: use curl or even fopen() to request the data.
you can do a man in the middle attack on the ssl connection an then sniff it but i don't know how important that is for you. i think i would analyze the sourcecode first here before starting to sniff the ssl connection unless you have a suitable setup at hand to do so.