Yes, body armor was experimented with - and even issued - by many armies in World War I. The most widely-known was the German "trench armor," called grabenpanzer or sappenpanzer, which consisted of a front-facing thick steel breastplate with additional plates (held on by straps) covering the abdomen; it was typically accompanied by a thicker steel plate that attached to the helmet. As many as half a million sets of these were issued, but they were heavy and generally only protected from pistols and bayonets.

There's a great book about that and various shields, mantlets, helmets, etc. called Helmets and Body Armor in Modern Warfare (1920), which is based on the Metropolitan Museum of Art in New York's collection.