So one of my regular received her donation in a white envelop. After the client left, she opened the envelop and to her surprise, a credit card authorizaton to pay, with signature, was included. So did she get robbed of her service or can she cash the payment at a bank?

she knew about it prior? did he ask about CC payment?  

Even if you could process this some how it creates a papertrail back to your real life identity... for an illegal actvity.   Sure n sh*t you'd get a charge back for fraud when his wife sees the  charge.  

Credit card authorizations are not negotiable instruments like cash, a check, or bearer binds.  
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She cannot get paid by a credit (or debit, or pre-paid) card without having a merchant account issued by a bank or an agent of a bank like Square (in the US).
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Getting this is problematic for providers as the contracts for these accounts specify that the "merchant" cannot accept payments for, or engage in, any illegal activity.    
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As such she risks exposing her "true" business to a financial institution which is obligated by card association (i.e.: bank) regulations and possibly state & federal law to shut down her account, seize her payment funds and report her to LE.  
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As to the guy who left the paperwork behind, he is either a criminal or an idiot. If he intended to get out of paying for services, he's a crook (even if the "service" might be illegal) and I suspect the credit card number and info provided is either stolen or fake. If he really thought she could "cash" it, he's an idiot because, assuming it was his real credit card info, she can pass/sell that info to any number of crooks who will generate false charges on the card up to and even beyond the credit limit.