Way to update a Mac to High Sierra or Mojave if OEM SSD is unavailable

In general, no. Apple doesn’t release tools to reverse an EFI update publicly. In general, these aren’t needed since the drivers for GPU and other functional aspects of running the system exist at the OS level which you can revert to freely. Also, EFI updates now arrive with normal OS updates, so most people get these updates when you apply normal updates and not needing to run a separate EFI update.

Technically, Apple service providers can get access to tools to revert firmware but that is more geared to when an update hangs as opposed to there being a known regression.

Known regresssions generally are resolved with a newer patch on top, always advancing EFI versions as opposed to reverting.

Now - hardware like the Mac Pro (Early 2009) may have other worries due to being classified Vintage and Obsolete products as defined by Apple: https://support.apple.com/en-us/HT201624

Also - there is a nice paper by some long time Mac administrators that work for a security firm now with research on EFI updates and what happens without a check and users being shown when they are down level:

• https://duo.com/blog/the-apple-of-your-efi-mac-firmware-security-research

It's getting a fair bit of mainstream coverage - but the headlines there are a bit click bait / overblown in my experience so far. The risk to home users is very low and EFI is generally only needed to exploit systems when they are already running 10.12.6 and have all updates applied and you can grant local physical access by a skilled technician of the hardware that's being attacked.

In specific, the Mac Pro 5,1 is listed as never having an update in the DuoSec research paper - so all this might be a moot point if all you care about is Mac Pro 5,1.

The EFI partition is used when the firmware is updated. I assume this EFI partition must be on the primary internal disk. Apple posted that firmware updates require a GUID partition scheme at the archived website Firmware updates for Intel-based Macs require a GUID partition scheme, although the site does not explicitly say this must be the primary drive. When a SDD or HDD is formatted to use a GUID partition scheme by the Disk Utility or diskutil command, the EFI partition is automatically created.

Basically, the firmware update is copied to the FAT32 formatted EFI partition. The firmware is then instructed to update itself. Basically, the same procedure occurs for firmware updates on my HP PC running Windows. So, this is not unique to Apple Macs.

Having to update the firmware to run High Sierra makes sense. The APFS is new and older Macs would need a firmware update before being able to recognize this filesystem and read the boot files.

You used to manually update the firmware. At some point, Apple decided to include firmware updates in the macOS installs. Apple posted this in the now archived website About EFI and SMC firmware updates for Intel-based Mac computers.