If a company receives a Phase III contract based on work done in a Phase I SBIR|STTR before they have been awarded the Phase II, is that company disqualified for applying for the Phase II award?

No, while a federal agency may enter into a Phase III SBIR agreement at any time with a Phase I awardee, no follow-on prohibition precludes the Phase III recipient from then submitting a proposal for the Phase II. The effort proposed must meet the intent of a Phase II and cannot duplicate the effort being performed under the Phase III.