The HSA custodian is indeed often a bank, but its information is highly unlikely to appear on your W-2 or anywhere else.
Do you have a debit card that was given to you by the custodian to pay for qualified medical expenses? If so, THAT is your HSA custodian (i.e., the Payer). Call that bank or institution and ask for the HSA department.
If that doesn't work, call the benefits person at your employer. Normally, he/she will know the name of the HSA custodian (since your company likely sends money to it).
P.S. it is likely that your 1099-SA is available online at the custodian's website. Do check.
