Since when do applications 'run' operating systems?
But a) original poster is talking about a web application; not a virtual machine or hypervisor and b) VMWare itself still requires an underlying operating system-- one could argue that this case is actually an operating system running another operating system with a thin adapter layer in between (i.e., the VMWare application).
VMware is a company, not a product - the product you're [probably?] referring to is ESX[1], which does not require an underlying OS - it's a Type 1 hypervisor (it runs on bare metal).