Oysters obtain the calcium carbonate for their shells by filtering it from the water. It is not obtained from a diet.
"Oyster larvae essentially “wrap” themselves around their few little individual molecules of calcium carbonate to get the shell started. They then add to their shell by filtering calcium out of the water."
This might interest you if you are interested in how oysters are farmed (if you can make it through the horrible formatting issues. They obviously didn't use plain text and instead copied and pasted from Microsoft Word into the site.
https://www.thekitchn.com/island-creek-oysters-grower-tour-192689
I guess it all comes down to how you define diet/eating versus just filtering. For me, filter feeders (which oysters are) are "eating" stuff from the water. Both to build their shell, as well as to build their protein/fat/carbs in the "meat" of the oyster. So then the question is really "When the meat is all gone and it is just a shell, does it matter what the oyster ate"? When it comes to if someone would consider it organic or not.
I suppose if someone accepted that ocean water and all that comes along with it is organic, then oysters are organic, both shell as well as meat. But if someone is worried about human wastewater and the things that come with that (prescription drugs both flushed whole, as well as excreted via urine and feces), then there are issues. Also runoff from agricultural fields, etc...