This is likely some callcenter-adjacent scam. They're probably doing this to hundreds of shops, and occasionally get one cave in and pay.
The products are all custom, e.g. your family's last name laser cut onto wood.
https://www.etsy.com/search?explicit=1&q=last+name+sign&ref=...
Your sister should contact the FBI, as it is unlikely that she is the scammers' only target, and this crime likely crosses state (or international) borders.
Actually the amounts cited are well below the FBI's interest threshold in any case.
And the amounts cited are not below the FBI's interest threshold; the whole point of my comment is that their are probably other victims, which makes this a larger crime than just the OP's system and probably involves sophisticated or organized criminals. This makes it exactly the kind of crime the FBI loves to investigate.
May you share a source for where you heard this information?
Cheap and scalable server time? No problem.
Communication with actual people? Not unless there's an exodus of customers (and even then you're not big enough to matter).
She runs an Etsy store with $1-2 million in annual sales, and her store keeps getting taken down by Etsy's automated copyright infringement system -- which keeps getting triggered by someone submitting fraudulent copyright claims and then immediately asking her to pay $5k/month in exchange for the person to stop submitting the infringement claims (in other words, she's being extorted).
Basically Etsy immediately takes down listings with no human review upon receiving a copyright complaint, which can be used by black hat scammers to extort stores into paying $5k-10k/month in exchange for the black hat to stop submitting fraudulent claims.
It's really astounding that companies build these automated systems that hurt their customers with no humans on standby to resolve these kind of edge cases (false positives).