Yes, you need to still comply with the requirements of Health Order 8, Intrastate Travel or Health Order 6, International and Interstate Travel. It is unknown whether vaccinated individuals might still be able to pass the virus on to others.
While the vaccines authorized by the FDA for COVID-19 are safe and highly effective, they have been tested only to find out whether they protect the person getting the vaccine from getting sick with COVID-19. We do not know yet how well they prevent the person from getting infected with the virus and passing it on, only how well they prevent the person from getting sick. We know that getting vaccine helps to protect you, but we do not know yet how much it helps to protect other people.
So, if you are traveling and you have gotten the vaccine, you are much less likely to get very sick with COVID-19. However, because you may still be able to catch the virus and give it to others, the requirements for quarantine and testing do not change. The vaccines do not affect your test results.
DHSS traveler information page: https://covid19.alaska.gov/travelers/
DHSS health guidance for Alaska travelers: