How does Google Voice access the microphone on the iOS through the browser? Apple hasn't approved the Google Voice app so Google created it as a web page.

My understanding is that the web browser doesn't (currently) have access to the hardware (such as the Microphone).

So, how does Google access the Mic through the browser?
It doesn't access the phone hardware directly. Instead, you enter the number and Google's servers call the number entered and simultaneously calls your iPhone on it's number. 

Merging those two calls happens in google's servers, putting both you and your recipient in communication through their hardware.