Since vSphere 2.5, there is a feature allowing you to capture screen shot of a running virtual machine. It’s not well publicized but you can find a short description with screenshotSupported (boolean) in the HostCapability data object. Thanks to Nikita for bringing this up in the comment of the vSphere Java API 2.1 GA post.
Indicates whether the screenshot retrival over https is supported for this host’s virtual machines. If true, a screenshot can be retrieved at the HTTPS relative path /screen?id=<managed object ID of virtual machine or snapshot>. If any of the optional parameters ‘top’, ‘left’, ‘bottom’, and ‘right’ is specified, the returned image will be cropped from the rectangle with upper left corner (left, top) and bottom right corner (right – 1, bottom – 1). These values default to the top, left, bottom and right edges of the image. The client must use an authenticated session with privilege VirtualMachine.Interact.ConsoleInteract on the requested virtual machine or, in the case of a snapshot, the virtual machine associated with that snapshot.
The managed object ID of virtual machine is the value of ManagedObjectReference, which can be easily found using MOB.
Once you have it, you can issue a URL as follows in any browser and get the screen shot in PNG format.