Charitable Donation

I think not, as the fee is for a service your client received.

I think so, because the fee is for a service the charity received.

Was it paid to the charity?  That makes an even better case.  

Isn't there a place in the software, for people claiming donated goods, to show mileage driven to drop the stuff off, which then calculates the deduction at the standard rate for charitable miles?  Or am I thinking of some other program?

Well, this settles it.  H&R Block says it's deductible. 

Out-of-pocket expenses

If you do volunteer work for a qualified organization, you can deduct:

• Mileage expenses if you use your car while doing volunteer work for a hospital or school
• Mileage expenses for miles you drive to and from a charity to drop off donated goods
• Parking fees, tolls, or public transportation expenses while doing volunteer work or dropping off donated goods

https://www.hrblock.com/tax-center/filing/adjustments-and-deductions/charitable-donations/

Yes, I know, that doesn't mention payments to third parties to get the stuff there.  But that limitation would clearly discriminate against the disabled, who don't drive their own cars. If they took a bus or taxi, it would still count, right?  And are you going to say the moving van only counts if they ride along with it?  Maybe it's not wheelchair accessible.