Are there any guidelines for what constitutes an “organic acid” regarding storage requirements?
@Jan is correct. 

Organic vs. inorganic is not the main concern for acid storage.  The strength and oxidizing ability of the acids is a much more serious issue.

Your institution/company may already have internal safety guidelines that you should follow.  If not, your country/province/state will have occupational health and safety regulations that you will need to respect.

You can find the American Chemical Society's safe storage guidelines here.