Using an API to change Accepts Email - Can this be done?

It is possible with the "update" method of the Constituent API. That documentation doesn't list all of the constituent fields because the list can be different for each organization. You can use the "listUserFields" API method to get a list of all of the fields enabled for your site. As another resource, consult the XML Schema definition for the constituent APIs at http://service.convio.net/xmlschema/crm.public.constituent.v1.xsd. This documents the API method response documents. For constituent records, the input parameters reflect the same structure as the "getUser" response. Because examination of that response record reveals an "accepts_email" field within an "email" field, we can conclude that the corresponding input parameter is "email.accepts_email" (using dot notation to separate the hierarchy of fields).