The Fix: You're the boss. First of all, you need to set the example by taking vacation. (I note that you're already starting to stockpile.) You need to do this by not moaning and groaning about what you'll miss when you're out. Any such moans and groans should be done privately.
You also need to address the stress level head on. As a group, tell your employees that they need time off. That it's healthy for them. That you will reward people who take time off. That it will increase their productivity if they are well rested and relaxed.
Also, make sure you remind people that vacation can be taken in small increments. Taking an afternoon off may do wonders for morale and won't be as complicated as being out of the office for a whole week.
Make sure that people aren't punished by taking time off. When you're making decisions on how to allocate work, or who to assign to projects, don't let people be punished when they take time off. This is actually harder than it sounds. It's too easy to fall into the trap of "well, Stephanie didn't produce as much as John did," because Stephanie was out of town for a week.