Start by talking about things that they are comfortable with and like to talk about, then segue into the interview topic.
THIS!
You start the conversation while you're setting up the lights, mics, etc. Transition into the actual interview after having already established a rapport.
Maybe there's an interesting knick-knack on their desk. There's got to be a story behind that. Maybe there's an interesting picture on their wall. There's definitely a story behind that. These things are where they are because the person who put them there likes to tell those stories.
But what if you're in a neutral space? Just start talking about regular everyday chit-chat. Gosh, the weather sure is nice lately. I miss the weather back home in Seattle. Wear did you grow up? Oh yeah, what was that like? Ask the kind of questions you'd ask someone in an elevator. Keep it impersonal at the outset. Establish rapport and get personal later.
There should be no official start of the interview. At some point, they might ask, "oh wait, has the interview started?" That's because you're sneaky. You just start a conversation, a normal conversation that we have on a regular basis. And then you sneakily transition from that normal everyday conversation into an on-camera interview. GOTCHA!
It's not just shy people who are difficult to film. ALL people are difficult to film. Put a camera in front of someone and they start to act different. Very few people are able to act natural when a camera is pointed at them. That's why we pay them millions of dollars to play pretend for a few weeks.
So, if you want someone to act natural when there is a camera pointed at them, start out by getting them to act natural without a camera pointed at them, and then slowly, incrementally, introduce the camera.
The interview should just be a continuation of the conversation that started long before the interview started.