The Short Answer: Your Credit Score Usually Doesn't Matter
Most banks and credit unions do not pull your traditional credit score when you apply for a standard checking account. Opening a checking account is not a lending decision, so your FICO score or VantageScore typically plays no role in the approval process.
What banks do check is your banking history, and the system they use for that is completely separate from the credit bureaus most consumers are familiar with. If you have a low credit score but a clean banking record, it can be useful to have no trouble opening a checking account at most institutions. If you have an excellent credit score but a history of bounced checks or unpaid bank fees, you could still be denied.
Understanding the difference between your credit report and your banking report is the key to navigating this process. They are two distinct systems, maintained by different companies, tracking different behavior.