Yes, you can still file a car accident claim. An arrest for drunk driving is a criminal proceeding. The criminal case is investigated by the police and prosecuted by the local, state, or federal government. The conviction results in financial penalties for the offense, costs for court proceedings, and other penalties including incarceration, probation, license suspension, and more.
A car accident claim is a civil lawsuit. Instead of seeking a conviction for a crime, the plaintiff in the case seeks to prove that the other party is liable for the accident and seeks compensation for the damages the liable party caused. It is not uncommon for a drunk driver to face a criminal conviction and civil liability for the same accident in separate court proceedings.