It is actually up to the Secret Service agents' discretion whether or not they want to wear a vest, as they can be bulky and hot.
This was a question that was brought up on a West Wing forum. Most of the answer is conjecture based on the character in the show, they also received a real email answer:
Wouldn't a secret service agent be wearing a bullet proof vest?
We were sent the following via email: "As someone with some experience working press advance for the President, I can tell you that it has been my experience that Secret Service agents do not wear vests. In fact, I don't think I have ever noticed anything other than they wear very nice suits (required to be black, charcoal or navy) and white shirts."
This is backed up in In the President's Secret Service by best selling author Ronald Kessler, who has a number of non-fiction books on the Secret Service, FBI, and CIA:
Since the attempts on Ford’s life, presidents have generally worn bulletproof vests at public events. They are currently Kevlar Type Three vests that will stop rounds from most handguns and rifles but not from more powerful weapons. Agents on the president’s and vice president’s details are now supposed to wear them at public events, but some agents prefer not to wear them. While the vests have been improved, they are uncomfortable and can make life unbearable on a hot day.