Does OSHA have jurisdiction over the Army?

By Executive Order the Army, and the other branches of the US armed forces, are directed to provide occupational safety and health at least as stringent as that of OSHA.

The executive branch administrative agencies (like OSHA) don't generally have authority on military installations, or in military operations. Typically there will be a military manual that essentually restates the civilian requirements, but is enforced by inspectors in the military. Of course, if the General has an issue and calls the local OSHA office an inspector would be willing do an inspection.