No, there is no penalty for paying with cash. However, businesses are required to make federal tax deposits via the Electronic Federal Tax Payment System (EFTPS) and face a 10 percent penalty if they don't. Penalty relief is potentially available on a case-by-case basis for taxpayers without access to a bank account based on reasonable cause. It is the taxpayer's responsibility to request a penalty reduction or an abatement.
Sources:
Publication 15 (2024)
Instructions for Form 8300 (12/2023)
§ 6050I. Returns relating to cash received in trade or business, etc.
Publication 1544 (9/2014)
IRM Part 4. Examining Process. Chapter 26. Bank Secrecy Act. Section 10. Form 8300 History and Law
Publication 583 (1/2021)