First, by law any person, paid or not, who sells paper pull-tabs for your organization is acting as an agent/employee for your organization. That makes your organization their boss—at least as far as selling paper pulltabs is concerned—and your organization has every right to insist that the required forms are properly completed.
Ultimately your organization is responsible for ensuring completion of the required forms. Because missing or incomplete game tracking records can result in citations, fines, or other sanctions, it may be better for your organization to discontinue gambling at a premises where employees refuse to do the required paperwork.