How do apply the Chicago Style referencing system using LibraOffice?

This is required for me to complete a degree essay.

I suppose you mean this. Since your question does not provide a lot of context (this is an understatement), don’t expect to receive informative answers.

The short advice is “learn LO Writer style management”: paragraph, character, frame and page.

If you’re looking for extensions providing already cooked styles, explore LO site. If you’re focusing only on citation and bibliography, explain yourself.

Additional information should be provided with an edit to your question, not an answer which is reserved for solutions. This site has no thread (only comments are poor approximations to a discussion). Also consider that the majority of us aren’t native English speaking people nor versed in North-American implicit typography usage.