What bank does TurboTax offer it's customers to use if they don't have a bank account

TurboTax does not have any bank that users of the software can use to have a tax refund deposited.

If you do not have a bank account or a debit card associated with a bank then, if you are eligible, you can request to have the refund deposited on the Turbo® Visa® Debit Card.

See this TurboTax support FAQ for the Turbo® Visa® Debit Card - https://ttlc.intuit.com/community/refunds/help/how-to-get-your-tax-refund-on-a-turbo-prepaid-visa-card/01/27310

The Turbo® Visa® Debit Card is provided by Green Dot Corporation and is issued by Green Dot Bank pursuant to a license from Visa U.S.A Inc. Green Dot Corporation is a member service provider for Green Dot Bank, Member FDIC. Visa is a registered trademark of Visa International Service Association. Green Dot is a registered trademark of Green Dot Corporation. Intuit and Turbo are trademarks and/or service marks of Intuit Inc.

Green Dot Bank operates under the following registered trade names: GoBank, Green Dot Bank and Bonneville Bank. All of these registered trade names are used by, and refer to, a single FDIC-insured bank, Green Dot Bank. Deposits under any of these trade names are deposits with Green Dot Bank and are aggregated for deposit insurance coverage.