The answer to your question depends strictly on state law, and on how the account was held. For example, a joint account usually becomes the property of the surviving account owner(s) immediately.
If an account is held only by an individual who dies, his or her estate becomes the owner. From a practical perspective, the account can't be used until someone is appointed by a court to represent the estate.
In some states, small-balance accounts of deceased individuals can be claimed after a statutory waiting period by the decedent's next of kin. The bank should be contacted for details as to documentation required, the length of the waiting period, the limits on the size of accounts that can be handled in this way, and indeed whether the state allows such a procedure at all.
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