No. The updated bivalent formula is in use only for COVID-19 booster doses, and not for initial vaccination. The best way to protect yourself from getting severely ill from COVID-19 is to get vaccinated. The CDC recommends that currently unvaccinated people get their primary series (the initial two doses of either Pfizer or Moderna, or one dose of the Novavax vaccine), and then wait at least two months to get the updated Pfizer or Moderna booster dose.
Added September 2, 2022