This is unanswerable at a US national level, but perhaps it is answerable at a state level. This document give the Washington state Phase 1A allocation guidelines. Whatever you plan to do, a first step is to request to be considered, here (a "please contact us" form). The government may contact you and ask more questions: here is the enrollment guide (questions you have to be able to answer). I will assume that the parties undertaking this scheme are complying with the law and providing truthful answers. There does not seem to be a question designed to figure out if the facility plans on hiring a thousand temporary workers to get a vaccine. They will ask about number of patients and staff, which would be a problem if the request for 300 doses suddenly became a request for 1,300 doses. There is also a triage system which assigns highest priority to e.g. bronchoscopy specialists in direct contact with infected patients. If people play by the rules, the Washington state system is designed to assure that only those at greatest risk in the health-care industry are eligible for Phase 1A doses.
Because I do not have a license, I cannot order vaccine from the Dept. of Health, and the DoH information makes it clear that you have to have licensed staff. DoH is constantly suspending credentials for unprofessional conduct, and the above scheme would very likely fall within the sphere of bad behavior that would allow then to pull one or more license. The written guidance is clear enough that your plan is contrary to their requirements that it's not reasonable to say "But we thought it would be okay". There may be even stronger reasons that emerge once you apply for vaccines and have to sign some paperwork.
It is also unclear if nursing homes will even qualify to directly order the vaccine given the storage requirements, instead Walgreens and CVS are administering doses. CVS gives a hint, that
CVS Health did not select the facilities. Rather, facilities selected
their preferred pharmacy partner through the CDC’s survey process. The
CDC then provided CVS Health with a list of facilities where onsite
clinics will be scheduled and conducted
One would have to contact CVS to see whether they require a facility to make any representations about their staff, i.e. do they ask "which staff have direct contact with patients?", to which a deliberately false answer could be fraud.