Buying tickets to see The Last Supper painting without taking a guided tour is fairly simple if you know the procedure and book well in advance (and preferably travel in the off-season).
Tickets are only available from the museum — either the official website or by phone. (The only reason to try the Last Supper Museum’s call center is to buy 6 to 9 tickets in one transaction. The available tickets by phone or from the website are exactly the same.)