Lots of people slurp clams from their shells. Since the clams are solid, that's not quite the same as drinking them. If you want to really drink a clam, run it through your blender or food processor.
It's easier to drink the juice that's inside the clam shell. When I steam clams I always capture the juice to use in clam sauce or to just sip as clam broth. Some people dip steamed clams into clam broth to get an extra-clammish flavor.
Clamato is a drink made of tomato juice with spices and clam broth and is used primarily as a mix for alcoholic beverages. The Bloody Caesar is a Bloody Mary with Clamato instead of tomato juice. Clamato is also added to beer in various beer cocktails, called Micheladas in Mexico and "beer 'n clam" in Western Canada.
The strangest line extensions of Clamato are not drinkable. You can get Clamato-flavored tortilla chips, and seasoned salt for coating the rims of classes.