So I tried using API version 11 and it still didn't work, but Customer Support (Thanks Russel!!) had a second solution which did work on our instance of Workfront.
The accessorID, accessorObjCode, and coreAction fields are in the accessRules collection, and can be updated on the object instead of the share endpoint.
URL = OPTASK/<issue ID>?fields=accessRules
Method = PUT
Body = {"accessRules":[{"accessorID":"<ID>","accessorObjCode":"USER","coreAction":"VIEW"}]}
Possible values for coreAction turned out to be "VIEW", "LIMITED_EDIT", and "EDIT" instead of view, contribute, and manage.
Again, thanks for your help!
Regards,
Kato