Historically, headshot photos were usually taken in a vertical format with one or both shoulders cropped out of the frame. The majority of the business headshots that I take today I are shot in a horizontal orientation. Horizontal means that the photographs are wider than they are tall. There are two reasons why I almost never shoot in vertical format for business photos.
1. Better cropping options: Headshots are used in multiple ways for modern marketing needs, so I rarely crop out shoulders or the top of someone's head for business photos. File sizes from today's cameras are much larger than we typically need or use. It's easy to crop a sharp image and it will still look great. You can always crop into a photo, but it's difficult to reconstruct a subject's shoulders with Photoshop if you need to create them from scratch.
A horizontally cropped photo looks great on banner images for social media and horizontal is best for hero images websites, and those can be easily cropped in for vertical or square photos for multiple uses.
2. More body inclusive: If you request vertical images, the shots will be half body photos. (Remember, we don't want to crop off shoulders, because designers hate that!) There may be members of your team who are not completely comfortable with showing half of their body in photos.
Ideally, team photos should be a positive experience for everyone in your group. We want to celebrate their work with your company, and help give your team a digital introduction to your clients.
Ultimately, deciding the orientation of the photos should be based on how you will use the finished images. Talk with your web designer and marketing team to discuss which will work best for your shots.