It appears that our issue was caused by stacking the laptops.  I forgot about this thread until the reply this morning, but I had found another forum and confirmed through support that this was the likely cause of the issue. It was explained to me that there are magnetic sensors in the top and bottom halves of the laptop that detect when the laptop lid is closed. When stacked, the sensor in the lid of the bottom laptop could trigger the sensor in the base of the top laptop and shut the laptop display off as if the lid were closed. We still occasionally stack laptops for convenience and desk space, but we make sure to offset the back hinges.  This has greatly reduced the number of times where we've seen the displays shut off while using the laptops.
Our organization has purchased close to 500 Latitude 5500, 5510, and 5520 laptops over the past 2 years. All 3 models have a very common issue that Dell support has never been able to diagnose, and I can't find any discussions of this issue online anywhere. The issue is that the built-in display will often just turn off while the computer is in use.  We usually see this when the laptop is first booted, but since I work at a help desk, we often see this issue when a user carries their laptop to our area for assistance with a different issue. When we lift the laptop off the desk, the display will usually come back on immediately, but sometimes, we need to tilt the laptop back and forth before the display comes back on.  Often, when we set the laptop back down on the desk, the display will go off again. It usually takes multiple cycles of lifting the laptop from the desk and setting it back down before the display will stay on. Sometimes, we have better luck if we set the laptop where the back end rests on a pencil where the keyboard tray stays slightly tilted toward the user. Usually, there is no detectible movement in the hinge when we are lifting and tilting the laptop, but we're still unsure whether the display going on and off is due to a sensor within the hinge or if there is some type of accelerometer in the laptop that is triggering this behavior. Another inconsistency is that this behavior sometimes will cause Windows to return to the lock screen as if the computer has resumed from sleep and requires a sign in. Other times, the display will come back on with whatever the user was looking at before the screen shut off.
Built-in display issues caused by movement