Coredumps are used for analysis and debugging if/when a ESX host "crashes" with a "kernel panic". This is visualized by a purple screen (often called PSOD, similar to Windows'es "Blue Screen Of Death").
VMWare ESX 5.5 and newer introduces the ability to perform coredumps to a file instead of a partition.
To configure this, you need access to the ESX host's CLI (either through vSphere Management Assistant (vMA), directly on the host through console or SSH, or some other method). For this to work, you need "root" access (or the equivalent of it through vMA).
Once logged on, take a directory listing of the VMFS datastores to determine on which datastor ...