Is there a way to update a Mac to High Sierra or Mojave if OEM SSD is unavailable?
The firmware updates are now included in macOS installs. Upgrading by installing High Sierra or Mojave to a USB external drive should cause the firmware to be updated. I have a external HDD with a APFS container partition. I installed High Sierra to this container using a 2011 iMac. I also installed Mojave in the same container (to a different volume) using a 2013 iMac. The firmware on both Mac was updated or I would not be able to boot from the external drive. (Although I can not boot Mojave on the 2011 iMac, which was expected.)