I’m new so don’t know much about WordPress but since nobody has responded to you, I’ll tell you what I did.
I added a text widget to my sidebar and put the web ring code in that. I didn’t use a title for the widget, just pasted the code in the text area. Worked fine for me. If there’s a better way to do it, somebody else will have to respond.
(I did first paste the code directly into the sidebar template file and then decided that was a dumb idea and took it out. It worked though.)
Good luck.