No. Many shoot using an iPhone. However your video should be high quality, 1080p, in focus, etc. I use a Nikon z6ii mirrorless camera but that is because I also use it for photography.
Modern webcams, such as the Logitech Brio, can also produce high quality video. Often the lighting matters more than the camera these days, as good cameras are available in lots of forms.
"Real" cameras offer more creative choices with different lenses, but you can produce good video with a phone or webcam too.
Whatever you use for video, don't neglect your audio quality. That's even more important.
I do use a DSLR for head shots, as I struggled to get the right quality from a webcam. I also found that it a webcam looked like a webcam video and I wanted a better quality result for the head shots. I did struggle with the audio quality so I invested in a Rode lavalier microphone which substantially improved audio quality.