Legally no one has to accept cash except for "debts, public and private". If a business is selling you something, they can stipulate any form of payment since it isn't a "debt".
What is a sales contract but the admittance of debts on both sides? The seller owes the buyer the good, and the buyer owes the seller the money. Everything is debt in contracts-land and it is quite fascinating. If you'd like, I can stipulate on how "no cash accepted" signs work even though you can force people to accept cash to settle debts.
I'm sure you could but it would involve setting up an in-person appointment at a specific location of their choosing. Also, expect to fill out a bunch of extra paperwork to show where the cash came from.
Legally no one has to accept cash except for "debts, public and private". If a business is selling you something, they can stipulate any form of payment since it isn't a "debt".