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.