She can contact any bank that offers HSA accounts. She would make money deposits directly (electronically or by check). She can deduct her contributions on her end of year tax return, so the tax savings will come in the form of a larger refund. rather than as reduced payroll taxes. Potentially, she or you could adjust your workplace W-4 forms to have less tax withheld each week in anticipation of the reduced taxes at the end of the year.
You may want to shop around. Different banks may charge different maintenance fees, and pay different interest amounts. I use an HSA that allows investment in a selection of mutual funds, and I have part of my money there to earn more than pennies of bank interest. I was not asked for proof of insurance and I don't think any bank will require proof of insurance, she will self-certify that she is eligible and that is between her and the IRS, the bank is not responsible for enforcing the insurance requirement.
