De Hawaii Pet Passport & Regulamentos de Importação (não é um site oficial, então leve-o com um grão de sal), sim, você pode levar seu cachorro para o Havaí. No entanto, parece haver requisitos bastante rigorosos antes e depois da chegada.
Hawaii has a five-day or less program for bringing cats, dogs and ferrets into the state. The program is designed to allow pets that meet certain qualifications to be released immediately upon their arrival at the Honolulu airport. You must start the process not less than 120 days prior to arrival in Hawaii in order to qualify to avoid quarantine.
Portanto, com cuidado, você pode pegar seu cachorro assim que chegar. Sem cuidado você pode encontrar o seu cão preso em quarentena durante a sua viagem. Observe também que parece que você deve voar para Honolulu, a menos que acordos especiais sejam feitos com antecedência - o que pode afetar seus planos de viagem.
You may enter Hawaii through Honolulu International Airport only between the hours of 8 am and 9 pm. Your pet will be taken from the airport to the quarantine center. You will pick up your pet at that location. Pet owners should be sure to arrange for their flights to arrive in Hawaii by 3:30 p.m. The process of clearing your pet generally takes one hour if your pet conforms to all requirements and has arrived on a direct flight. Otherwise, the process can take up to 4-5 hours. Pets not cleared by 4:30 p.m. will be held overnight at the Airport Animal Quarantine Holding Facility.
You may also enter through the neighbor Island airports of Kahului, Kona, or Lihue but special arrangements must be made in advance. Note that not all airlines participate in this program. All requirements for entering Hawaii must be met and documentation must be submitted at least 30 days prior to import.
Quanto à questão da companhia aérea, a resposta é sim em geral, mas isso depende da companhia aérea em questão.