We have a company-specific Stack Overflow site for folks to ask questions with the company. It has a *.stackoverflow.com domain/URL.
Would I be able to use the Stack Exchange API to access the questions, etc. from that site?
I'd like to create a tool to get insights from it and share it internally. If not, are there any other options available?
Are you using Stack Overflow Enterprise? If so, API support was promised for right about nowThere's a chance that ETA might slip. ;).
Otherwise, how exactly is this company-specific site configured? If it's not either in this list of sites, or visible to the general public on regular Stack Overflow, then you have no API support for now. Contact your account manager at Stack Exchange for more info.
