For whatever it is worth, I added 4 more NICs to an ESXi 6.5 VM running on VMware Workstation and added the four NICs to the same vSwitch. Chances are the HP card you are looking at will be using the same e1000 driver. The DID 100F and SSID 0750 doesn't agree with the VMware Compatibility Guide values of 10d3 and 07d0, respectively for the virtual Intel 82574L.
This is what appears on the VMkernel device manager log.
2017-05-30T07:50:29Z vmkdevmgr[65906]: ADD event for bus=pci addr=m01000208 id=8086100f15ad0750020000.
2017-05-30T07:50:29Z vmkdevmgr[65906]: Found driver e1000 for device bus=pci addr=m01000208 id=8086100f15ad0750020000.
2017-05-30T07:50:29Z vmkdevmgr[65906]: Binding to vmklinux for driver e1000
2017-05-30T07:50:29Z vmkdevmgr[65906]: ADD event for bus=pci addr=m01000210 id=1274137112741371040100.
2017-05-30T07:50:29Z vmkdevmgr[65906]: ADD event for bus=pci addr=m01000218 id=15ad077015ad07700c0320.
2017-05-30T07:50:29Z vmkdevmgr[65906]: Found driver vmkusb for device bus=pci addr=m01000218 id=15ad077015ad07700c0320.
2017-05-30T07:50:29Z vmkdevmgr[65906]: ADD event for bus=pci addr=m02000228 id=8086100f15ad0750020000.
2017-05-30T07:50:29Z vmkdevmgr[65906]: Found driver e1000 for device bus=pci addr=m02000228 id=8086100f15ad0750020000.
2017-05-30T07:50:29Z vmkdevmgr[65906]: Binding to vmklinux for driver e1000
2017-05-30T07:50:29Z vmkdevmgr[65906]: ADD event for bus=pci addr=m03000230 id=8086100f15ad0750020000.
2017-05-30T07:50:29Z vmkdevmgr[65906]: Found driver e1000 for device bus=pci addr=m03000230 id=8086100f15ad0750020000.
2017-05-30T07:50:29Z vmkdevmgr[65906]: Binding to vmklinux for driver e1000
2017-05-30T07:50:29Z vmkdevmgr[65906]: ADD event for bus=pci addr=m04000238 id=8086100f15ad0750020000.
2017-05-30T07:50:29Z vmkdevmgr[65906]: Found driver e1000 for device bus=pci addr=m04000238 id=8086100f15ad0750020000.
2017-05-30T07:50:29Z vmkdevmgr[65906]: Binding to vmklinux for driver e1000
2017-05-30T07:50:29Z vmkdevmgr[65906]: ADD event for bus=pci addr=m05000240 id=8086100f15ad0750020000.
2017-05-30T07:50:29Z vmkdevmgr[65906]: Found driver e1000 for device bus=pci addr=m05000240 id=8086100f15ad0750020000.
2017-05-30T07:50:29Z vmkdevmgr[65906]: Binding to vmklinux for driver e1000
And this is a partial screenshot of the vSwitch configuration. Your intentions for the 4 NIC ports might be different, but I think the gist of it is there.
