No there was no specific uniform for them. Stormtroopers were at first ad-hoc unit created locally in many places by various German commanders (Rohr being the most famous leading an experimental pioneer unit testing new weapons and tactics proving the concept to the high command) and the idea moved up the chain and policies were established to train more of them. In October 1916 all German armies on the West front were ordered to form them. But crucially, this was part of an effort in reforming the entire German army along stormtrooper lines, not to create a few elite units.

Thus Stormtroopers started out in normal uniform and mostly remained so. The one thing they had to distinguish themselves were their first adaption of the iconic Stahlhelm in 1916. They also sometimes personalized their standard uniform with patches on knees and elbow to strengthen them and getting other leg wear. These were mostly practical changes and regular officers didn't always appreciate these personal initiatives.

The main differences existed in how stormtroopers were equipped. Ie they had more weapons with more firepower and portable equipment than a normal rifelman had access to.

As always your source of uniform porn is Osprey, Warrior 012 - German Stormtrooper 1914-1918, by Ian Drury and Gerry Embleton in this case.