Can the constitutionality of a law in the United States be challenged other than through the Supreme Court?
In the United States, the constitutionality of a law can be challenged in any court. Unless and until the court is reversed by a higher court, then the law will be considered unconstitutional and invalid.
A jury can also theoretically nullify a law for any reason, including unconstitutionality, however, the jury's finding will only hold for that one case that they are trying.
It is not true that a law's constitutionality may be challenged in any court. Rather:
State Laws and the Federal Constitution
Federal courts may hear cases concerning state laws if the issue is whether the state law violates the federal Constitution. Suppose a state law forbids slaughtering animals outside of certain limited areas. A neighborhood association brings a case in state court against a defendant who sacrifices goats in his backyard. When the court issues an order (called an injunction) forbidding the defendant from further sacrifices, the defendant challenges the state law in federal court as an unconstitutional infringement of his religious freedom.
Some kinds of conduct are illegal under both federal and state laws. For example, federal laws prohibit employment discrimination, and the states have added their own laws which also forbid employment discrimination. A person can go to federal or state court to bring a case under the federal law or both the federal and state laws. A state-law-only case can be brought only in state court.