The idea is that the society collects souvenirs. Not commercial rubbish. They must be things that are not for sale, are of little intrinsic value, can be replaced with little inconvenience by whoever you got it from, and which are completely characteristic of the place you visit. Start with monogrammed ashtrays from pubs and restaurants. Move up to stationary saying 'From the desk of ...' from the Prime Minister or President of your choice. Or their nameplates from major summit conferences. Flags from outside UN headquarters, the Barleymont EC headquarters building (or a sample of the asbestos from the walls that makes the place uninhabitable) would be good. Taking The Mona Lisa from the Louvre is clearly a violation of the rules, but how about the rope that hangs in front of it?
Nuclear power stations offer unusual opportunities and challenges, now that they are open for public tours as part of their image-making campaigns. A few kilograms of unenriched Uranium is not actually very dangerous and in today's' world alarmingly easy to replace, and it packs into a surprisingly small volume,. But someone might get the wrong idea if you are caught. Anyway a large sign saying 'Sellafield: trespassers will be lucky to get out intact' would be much more characteristic.
The most ambitious one I have thought of so far is 50,000 cubic feet of helium from inside the Goodyear Blimp, but no doubt you can have your own ideas.
The key to success is in the rules. This must not be mere theft or vandalism. It must be part of a genuine tourist visit, bringing back something that will remind you afterwards of the place and time, and not just because you had to escape the police by driving at 95mph across the border with a three hundred feet of stairwell from the Eiffel Tower bolted to the roof of your car. Perhaps the ultimate would be bringing back a few cubic centimeters of Genuine Lunar Vacuum (but only if you had collected it yourself).
Anyone want to be membership secretary?