After looking on an online dictionary I figured out that we can add する to 表紙 meaning "To appear on the cover". However, after talking with a native Japanese speaker, she told me that this verb doesn't exist and they use another expression to mean the same thing. So my question is, does 表紙する really exist and if yes, how can I use it?
It doesn't seem to be an established construction. I've never seen it before, and when I looked for examples online, I was only able to find conversion errors where it seems people intended to write 表示する. I was unable to find any real world examples.
JMDICT, which is the dictionary used by the website you linked to, is a project maintained by a number of different volunteers and can be expected to contain some mistakes. In this case, my guess is that someone did run across 表紙する before adding it to the dictionary, but it was a conversion error rather than a legitimate example.
All of the other dictionaries I consulted listed 表紙 only as a noun.