Answer:
The risk is that the maker of the check has up to one year (timeframes vary by state uniform commercial code) to claim that a check as been altered and the payee information washed, removed, etc. If you are unable to collect from your customer on a deposit or you've cashed the check for a non-customer, then you stand to lose the amount of the check.
You have no way of honoring your presentment warranty under the uniform commercial code that the right person received the proceeds from the check. I would not accept a check without a payee.