You are correct that there are three exceptions to the job posting requirements. The mandatory job listing requirement set forth in the VEVRAA statute requires covered federal contractors to list "all employment openings" with the exception of 1) executive and senior management positions; 2) positions filled internally; and 3) positions lasting three days or less. If your position does not fall under any of the three exemptions above, then OFCCP will expect you to list this position with the appropriate Employment Service Delivery System or ESDS (also referred to as state job bank) where the job is located. The OFCCP does not have specific requirements for interns, however, interns who are paid by an organization and whose work is controlled by that organization, regardless of number of hours employed or nature of the work, would typically be considered employees. Based upon these factors, it would be in your best interest to ensure these positions are posted - either by you, or your third party.
You can use this OFCCP audit checklist to ensure you're doing what is required to maintain OFCCP's regulations including VEVRAA, Section 503, and EO 11246. Or request a demo to streamline your compliance and recruiting efforts.