If you think writing out your answer will help you come to the right solution, then ask for permission to use pen and paper. Make sure that the interviewer agrees before moving forward with it. More importantly, be prepared with your notebook and pen next to you when you sit down for the interview, so you do not have to get up in the middle of the video conference. This can be a sign that you are not prepared, which can have a negative effect on your score.
Since you will be writing on paper that they can’t see (as opposed to a whiteboard), you need to be extra careful about explaining what you are doing and why. You can even hold up the paper every so often to show them. Make sure you explain your process each step of the way.