Can an adjuster refuse a claim when their driver was issued a ticket at the scene of an accident?

The broad answer is "No", if only because a ticket is merely an accusation, not a determination of guilt or liability.

But the issue gets more complicated depending upon the coverage involved. For example, collision coverage is a first-party coverage intended to pay for the physical damage to ones own car. If the claim is otherwise covered (meaning essentially that the insured has done everything required by the policy to assert the claim), the fact that the insured was given a ticket, and even if he convicted of the offense, will not ordinarily bar recovery.

This is in contrast to another kind of first-party insurance, uninsured motorist coverage. It is designed to pay the same sort of damages to a person injured in a collision which the other driver or owner's liability insurance would have paid if that person had bodily injury liability coverage (such as damages for a permanent injury, pain and suffering, disfigurement, etc.). Stated otherwise, uninsured motorist coverage takes the place of the at-fault driver's/owner's bodily injury liability coverage when he/she does not have that coverage.

Normally, the payment of uninsured motorist coverage benefits to an injured party will follow the same rules of negligence law as a claim directly against the at-fault driver, and thereby, principles of comparative or contributory negligence usually apply. Therefore, if the claimant is cited for a moving violation related to the collision and convicted of the violation, that finding may be interpreted as some degree of negligence, which may, in turn, have a bearing upon recovery from the insurer.