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The reason they probably don't accept cash is to prevent any of it mysteriously disappearing, and checks can bounce. Although they are losing a few percent on the credit card transaction, they probably consider it worth it since they don't have to worry about cash or chase bad checks (which can be a major hassle and you still may not collect).

I think they can dictate how they are paid as long as they tell you up front. If they did not tell you and then you showed up with cash and they refused it, I think you could make a case that you offered to pay for the services but they refused payment. After all, what if you showed up to pay and they said "we only accept gold coins or goats as payment". I think you could possibly take something like this to court and win. This may be why business usually have large easily seen signs displayed saying either they don't take cash or bills over $20, etc.

I once lived in an apartment long ago (last century) where they suddenly stopped taking cash and I told the lady OK, I offered to pay and you refused, so I guess the rent is free this month. I returned to my apartment and about an hour later she came to my door to say she could take the cash. I wondered who she talked to and what they told her. She explained that they didn't want to keep cash in the office any more because they were held up once. The next day, all the occupants got notes on their door stating that they would no longer accept cash and the reason why. I don’t know how the business in question would handle it if the customer said they did not have a credit card. I guess they would accept a cashiers check or money order, or turn the customer away.

Of course it could also be that they are secretly recording the card information and selling it to other crooks.