Madison does have departments just like any flagship and most large stores; however, a scarf specialist/homewares manager/etc. can sell you bags.
Otherwise it's not like FSH or any Paris store, at all of which tourists can only make a leather appointment via lottery, and can get bags without a profile or "prespend".
It is like other US flagships in large cities, where locals with a big profile are vastly preferred, but if you hit it off/buy something meaningful, they have stock and therefore you have a shot. As with any such situation, my personal advice would be to buy only something you want, not as a means to a bag, and only if you really think buying it at your home store is not a better path.
For non-Birkin, Kelly, Constance, KTG/CTG, or very coveted bags like mini Lindy - bags other than those are a bit easier as a walk-in.