Which word processors and websites support vertical writing of Chinese?

Regarding #2, you can try installing Stylebot or a similar browser extension, then adding the following custom styles:

body * {
  writing-mode: tb-rl;
  text-orientation: upright;
}

#stylebot-container * {
    writing-mode: lr-tb !important;
}

Layout will be anywhere from a bit messed up to massively messed up, depending on the site, but the text will be vertical. Not sure exactly why you'd want to do this, but there you go.

Is there a convenient (preferably free of charge) word processor if I wish to type a Chinese document vertically?

Microsoft word support this function. Click layout -> text direction -> vertical.

TextEdit

Format/ Make layout vertical