Can I sue a non- profit organization if I injured myself within their store?

If the injury you received while on the premises of someone's  home or place of business is actionable, so long as you can prove negligence by the opposing party.  Whether the organization is a for-profit or not-for-profit entity is irrelevant.  Of course, depending on the facts, they may argue that you may have been contributorily negligent.  Thus, you should speak to an attorney to give more details about your specific case.