Yes, of course it's legal to petition the US Supreme Court. When discussing Supreme Court cases, it's important to realize "petition" means to file an appeal of a case already decided by the lower courts. It does not mean to send a signature petition in protest or support of a decision. You are within your legal rights to send such a document, but the Court is not required to read it.
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US Supreme Court cases are initiated when a minimum of four justices agree to grant a petition for writ of certiorari.
Petition for a writ of certiorari For more information, see Related Questions, below.
The US Supreme Court is the highest court in the US. Each state has its own Supreme Court, but the US Supreme Court is the end of the line.
A writ of ceritorari is filed as a petition to the US Supreme Court for them to take a specific case.
The only state courts eligible to have cases reviewed by the US Supreme Court are the state supreme courts or court of appeals immediately under the state supreme court (if that court declines review), because a case must exhaust all appellate options before any party can petition the US Supreme Court for a writ of certiorari. State supreme courts can avoid having their cases reviewed by never accepting a case that involves questions of federal or constitutional law. Barring that, they could pay careful attention to case law and not make a ruling in conflict with an established precedent. Barring that, they can only hope whoever loses the state supreme court appeal doesn't petition the US Supreme Court for a writ of certiorari or, if they do, hope that the case isn't sufficiently compelling that it attracts four of the Justices' interest.