Generally no. If your computers are all Dell, though, you might be able to get some information (maybe the serial number?) for the PC itself.
The monitor, if it supports VESA EDID (DDC, EDID, EEDID), may also include a 32 bit serial number - which may or may not have any relation to the serial number printed on the monitor's label. You may be able to access this through the display driver - Windows has access to portions of it (to display monitor resolution and timing) so I expect the manufacturer/model/serial number is stashed somewhere as well.
However, making such a program that would work across all systems and monitors would likely be much more work than simply going to each station and recording it, unless all the systems have the same hardware.
Good luck!