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I am very sorry to hear about your family's loss.
Unless your sister's account had a payable on death clause (POD) that names him as the beneficiary, he cannot (or should not) have any access to that account and cannot close it without first filing for a probate action.
When someone passes away, then their estate has to be distributed. The problem is that without probate - with assets such as titled property or bank accounts (those that do not have a pay on death clause) - this is hard to do. This is because you cannot switch over the assets without an order from the probate court, and simply a Certificate of Death will not do. A Certificate of Death simply states that someone has passed on, but does not give you the right to really do anything in the deceased's name.
So one files probate. Once probate is filed, the Executor of the estate gets something called a Letter of Testament/Administration (hereinafter "Letter"). This Letter will allow the Executor to switch over the assets from the deceased individual to whoever will own the property. It is like a "Power of Attorney," but from the Court. Without that Letter, there is no way to transfer titled property and switch the assets into the heirs' names.
If there was a Will, the beneficiaries are decided per the Will. If there was no will, the beneficiaries are decided by default succession law of the state.
Now - assuming the bank account has no POD clause, he needs the Letters of Testament/Administrator to access it. This would involve filing for probate. However, her estate may be liable for small estate administration which is GENERALLY DONE VIA MAIL and is much simpler. See HERE and HERE. This gives the Letter of Testament/Administrator quicker and easier.
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