Can two independent organizations merge?

Yes. Assuming that they are both corporations, it is not unusual for two charities to merge. (See Ready Reference Pages: “Mergers and Acquisitions Can Take Many Forms” and “Mergers and Affiliations Require 'Due Diligence'.”)  Some states have liberalized their laws so that corporations can merge with limited liability companies or other entities so you should check the relevant state laws if both organizations are not corporations.