It's going to be a bandwidth hog. I'm assuming a gym is not going to have oodles of bandwidth to work with so hosting the video directly is out. Sending data to a third party service to distribute your video may be possible but will still use up a lot of bandwidth on a single stream. I've seen several places that will do this for your audio but haven't looked for video.
Your best bet is to record and cut a movie. iMovie can do it on the cheap but is limited in extra features. You can then place your movie on a site like youtube or vimeo and then iframe it into your website.
A recent iPhone or iPod camera can record decent video and so can most of the cameras sitting at for sale at Costco including DSLR type cameras.
If you're going to shooting somebody talking, the issue is going to be background noise and directional sound pickup. Most cameras do poorly without a directional mic or external mic when it comes to recording sound. In an enclosed quiet room it will not be noticed much. In a crowded room you will notice it. When you do decide on a camera, one with a mic jack will come in handy. A wireless wearable mic will also come in handy.
It may be tempting to go with a $25 tripod. A tripod you can be happy with should cost 5 to 10 times that and more. A bad tripod can ruin your shot.