Not unless the state legislature specifically grants that power.
Article II Section 1: Each State shall appoint, in such a Manner as the Legislature thereof may direct, a Number of Electors . . .
Legislative power over Presidential elector appointment is thus plenary. That is why there are fifty various statutes regarding elector appointment. I have not looked specifically at AL statutes and don't care enough to take the time, but unless AL specifically provides for judicial review of qualifications, the courts lack jurisdiction.
Also, courts love precedent. Allowing courts to toss election results is a bad habit to develop in a representative republic.
Several states have this power granted through specific legislation. They have the power to decide who can and can’t be on their ballots, many with specific requirements to meet constitutional eligibility. The courts, through the Constitution of the United States, have the enumerated power to resolve controversies arising under the Constitution.