Let's say that I have a recruiter's email but I don't have her phone number but I wanted to reach her via phone before I email her, is there any way I can get her extension number from the company? Can I call the company and ask for her extension number?
Just call the company and ask to be connected to that person. And if you talk to the person and want to call again later ask if she can give you her extension number for next time. Maybe she will give it to you, maybe she will ask you just to call the switchboard.