I would like to have a mercurial repository on my website so that I can push/pull as I make updates to it, but I do not have SSH access, only HTTP or FTP.
Can this be done?
I suspect no, since I would not be able to run hg on the server, so I would only be able to clone it.
You're correct. If you can't install software, you could use the static-http option, but it only supports cloning/pulling. See the comparison of publishing mechanisms.
I'm also using only FTP to update my site, but I keep all the files in a local Mercurial repo (you could use a repository on bitbucket - they have a free plan that includes 1 private repo and 1 GB of available space). When you need to add/update something, apply the changes locally, update the Mercurial repo, and then use FTP to update the website.
Yeah, this can definitely be done. You don't need ssh access to install mercurial or to access it. You compile it on your own system and then FTP up the resulting file. The only real requirement is that the website to which you upload it allow CGI applications to run. Without that you're limited to the static-http repositories, which don't support pushing.