Manycam is free; you can use that as the streaming client. Definitely use a webcam, there are several that can do 1080p, although 720p is sufficient for a good quality image. Many have built-in mics that, in a setting that you describe, should work well. If the environment around your client is noisy, then a separate mic is in order. Manycam handles both the audio and video.
All this requires a decent quality laptop preferably with USB 3.0. USB 2 will work, but you may not be able to get to 720p. You’ll need a remote control such as Team Viewer to help set up remotely.
Of course, an internet connection, with the laptop connected by ethernet. Wireless may cause problems and if your client is not a comfortable with tech, skip using wi-fi.
