On the newer Dell systems you can find out the firmware version by connecting to the iDRAC (Dell's BMC), it has all that information as well. This is assuming you purchased the NIC from Dell and it has all the Dell secret sauce in it to provide this information to the BMC.
No we did not purchase the NIC from Dell. Is there anyway I can find the firmware from the ESX? or while bootup?
That makes it a bit tougher, but not impossible. The tracking # of the EEPROM is stored in words 0x2D and 0x2E. You can use ethtool to dump these words:
ethtool –e vmnic4 offset 2d length 1
ethtool –e vmnic4 offset 2e length 1
We can then use that to associate with a specific version. What information are you trying to associate with a version?