Who in the fraternity would be prosecuted if this became an issue?
A lot of people could be held liable for this, including people who are not even in the fraternity. Anyone who has knowledge of the machine or the fact that it was possible for minors to access alcohol through it could technically be held liable if a prosecutor wanted to make that case. Presumably the building is owned by someone else and just leased out to fraternity members, and they very well could be held liable for sale to minors also.
Would the machine be safer if it just accepted cash (so that no electronic paper trail was created), with a big warning sign WINK WINK that anyone under the age of 21 was strictly prohibited from purchasing from it?
No. Payment method is irrelevant here. There are a number of states that legally allow vending machines to sell alcohol, but vendors are required to verify the age of any person accessing them and ensure that those cards aren't being used by people not authorized. What you describe is an extremely relaxed environment where admittedly no one is attempting to verify identities.
The "accepting cash" scenario is no different than a liquor store selling alcohol to anyone that comes in just because they're willing to pay with cash instead of a credit card. Sales to minors laws are not "as long as you warn them, you're safe" laws. They require vendors to actively check IDs and ensure that alcohol is not landing in the hands of minors. Accepting cash just to erase the evidence doesn't meet that burden.
There are a lot of legal troubles with the situation that could get a lot of people charged with multiple offenses. That you have underage fraternity members living there suggests you should not have alcohol readily accessible in the house at all, as most state laws expressly forbid providing access to alcohol, not just serving or selling. Them being there provides access to it, even if it's just a case in the fridge with a note on it.
Not to mention, you technically cannot sell alcohol, as I highly doubt your fraternity has a liquor license to be able to do so. There's a big difference between asking everyone to pitch in to buy the case versus actively selling individual cans through a vending machine. The vending machine itself is violating liquor laws in your state merely by existing.