My understanding is that anyone who's not a party to the action can submit an unsolicited letter (called an amicus curae brief)
Amicus briefs are filed by persons who are not parties in the case, but they are not the equivalent of unsolicited letters. Generally, it is not the case that anyone can file an amicus brief. Parties wishing to file such briefs usually need permission from the court. It appears the California court system follows this rule.
CA Court Rule 8.200(c):
(c) Amicus curiae briefs
(1)
Within 14 days after the last appellant's reply brief is filed or could have been filed under rule 8.212, whichever is earlier, any person or entity may serve and file an application for permission of the presiding justice to file an amicus curiae brief. For good cause, the presiding justice may allow later filing.
So applying the CA rule, not only would the letter presumably not be a proper amicus brief, but it would be out of time. An application for permission to file an amicus brief would appear on the docket if one were filed. I have never heard of a court allowing someone who is not a lawyer to file an amicus brief. The standard is generally that it is able to help the court decide the case, so filing such a brief requires discsussion of the law (usually some nuance of the law that the parties themselves are unable to delve into).