If I were to purchase the Map Stamps Collection, would I need eighteen inventory slots?
Map Stamps do not take up any inventory space. However, the first time you buy a map stamp, you receive the World Traveler's Hat.
The World Traveler's Hat will have a falling stamp particle effect on any map you own a stamp for.
Note that only user contributed maps that are now shipped with the game have stamps, of which there are (as of 2011/11/17) 18: Turbine, Egypt, Fastlane, Steel, Watchtower, Harvest, Junction, Mountain Lab, Mann Manor, Yukon, Freight, Hoodoo, Offblast, Coldfront, Frontier, Lakeside, Nightfall, Gullywash.
Your hat also has ranks based on the number of stamps you own for a specific map. This rank changes for every 25 stamps you own for a map up until 200. Again, these are tracked separately for each map.