When an exempt employee’s time is changed based on a punch system, does this make them non-exempt?

Yes, it does make them non-exempt; it also makes them non-salaried (i.e. they are now paid on an hourly basis). The law is very clear that to be salaried, an employee's pay is not based on the hours he or she works. (One exception: if he/she misses an entire day, they are not paid that days' salary unless they use paid time off.) If the employee works less than normal one day, they are still paid their full daily salary; if they work more than normal, they still only get their normal salary. To adjust pay by time worked--and to use vacation time to "cover" missing a punch in or being late or leaving early is to adjust their pay by time worked, since the employee has to use paid time off to make up for time missed--is to render the employee hourly, not salary, and therefore also give up the exemption from overtime.
An employer is free to demote, suspend, even terminate a salaried employee for not following policy or for missing time at work: that is legal under "employment at will," which is the law of the country unless the employee has a written employment contract to the contrary. But they can't base the employee's pay on their work time, and making them use PTO up for missed time is basing the pay on time worked.
Note that the employer may track salaried employee time for other purposes: to bill projects or clients; to manage workflow; to determine eligibility for or amount of bonuses; etc. But not for base pay.