I work at a non-profit, and the person in charge of the website has left. I have taught myself how to edit the website online through our web log in, but can't figure out how to edit the theme. All I need to do is add a logo to the theme (we use the Zen subtheme). Is this something I can even do online, or do I need to set up a server shell or server test site on my desktop?
Go to /admin/appearance/settings/YOUR_THEME_NAME and scroll to the bottom and find the logo field and upload your custom logo
See the following for a nice visual walkthrough.