Vagrant uses virtualbox for all its tasks. But it supports writing the vmware vmdk file format (virtualbox supports this). It seems to be possible to deploy those image files to EC2 and now I'm asking myself what does prevent it from running in a vmware environment when the image file has the right format already. EC2 does not run virtualbox either.
There are few different vmware disk image formats. Most probably you will need to convert the ESX file you have for deployment on servers to something else.
From the vmware-vdiskmanager for VMware workstation:
Disk types: 0 : single growable virtual disk 1 : growable virtual disk split in 2GB files 2 : preallocated virtual disk 3 : preallocated virtual disk split in 2GB files 4 : preallocated ESX-type virtual disk 5 : compressed disk optimized for streaming 6 : thin provisioned virtual disk - ESX 3.x and above
I would guess that the files you have are type 4 or 6.
It seems to me this ought to work, for the most part.
One issue is that Vagrant requires the VirtualBox guest extension kernel modules, which allow the guest OS to mount directories on the host (among other things). I suppose it's possible that those kernel modules will cause problems if you try to run on a non-VirtualBox VM.
But, try it. I think it's likely to work.
That said, VirtualBox is wonderful -- I don't see much reason to bother with VMWare Workstation.