I am the trustee in her will.
I'm going to nit pick your terminology because it's important.
A trustee is a person who administrates a trust. Did your mother have a trust in addition to her will or just a will?
The person who administers a decedent's estate pursuant to a will is called an executor or administrator or personal representative. My guess is that your mother's will nominates you to serve in that capacity.
I am unsure of how to go about claiming what’s in her bank account. She did her banking online. Can anyone give me some advice?
It's pretty early in the process. Can you give a rough estimate of the value of her estate? Did she own her home? Do you have all of her bank account information? Just one account or multiple accounts? Just one bank? Does the bank have an office near you? Did your mother receive social security deposits into her bank account? Was your mother married? Are there other living children and/or children of predeceased children?
How you answer these questions is important, so please answer each of them.