This is an interesting question because it is usually asked the other way around: may an organization refuse cash and require a check or credit card payment.  Many organizations, particularly during the COVID era, do not want to handle pieces of paper of unknown circulation.  Many ask for payment only by checks or credit cards. Although a few states require vendors to accept cash, there is no requirement under federal law to accept cash, although it is legal tender for payment of obligations if the recipient is willing to accept it. 

The U.S. Treasury Department says it is up to the vendor to decide on the means of payment it will accept for its transactions, so it seems that the organization can set its own rules for payment. If you are concerned with “leakage” from the gate receipts and improper private gain, these are issues that apply to any cash-based business. The concern might be less for a charity (although not insiders who might syphon off funds) since a tax-exempt entity has no taxes to avoid by failing to report income. The internal control over the accounting may be an issue, but is not a question you asked.

It doesn’t seem to make a great deal of economic sense to turn away paying customers (unless your neighbors have a reputation for bouncing checks on accounts with insufficient funds), but it appears that they have the right to set whatever policy they wish. I am not aware of any IRS rule on the means for accepting payment that would affect its tax-exempt status.