Whoever is the named executor needs letters of administration from the local surrogate court. You need to file a death certificate and the original will, and an application for the letters of administration. No one can collect from an estate without these letters.
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You might find my legal guide on Estate litigation (without a will) helpful.
Mr. Sarno is licensed to practice law in NJ and NY. His response here is not legal advice and does not create an attorney/ client relationship. The response is in the form of legal education and is intended to provide general information about the matter in question. Many times the questioner may leave out details which would make the reply unsuitable. Mr. Sarno strongly advises the questioner to confer with an attorney in their own state to acquire more information.
You likely need to do something to get the authority to access the account. Given the small amunt involved you may be eligible for one of NY's abbreviated procedures. I recommend talking to the probate clerk in your local Surrogate's office to see if they are helpful in determining what is the easiest course of action for you.
Very truly yours,
NOTE: This answer was prepared for educational purposes only. By using this site you understand and agree that there is no attorney client relationship or confidentiality between you and the attorney responding. This site should not be used as a substitute for competent legal advice from a licensed attorney that practices in the subject area in your jurisdiction and with whom you have an attorney client relationship. The law changes frequently and varies from jurisdiction to jurisdiction. The information and materials provided are general in nature, and may not apply to a specific factual or legal circumstance described in the question or omitted from the question.
If your mother's assets total less than $30,000, then you do not need to probate the will. If her assets total less than $15,000, you may not need to go to court at all. I have provided a link with more information on settling a small estate.