Last time I updated a HPE drive was 6 years ago, but downloading the Windows EXE payload and running it let my update my HPE Intel S3520 directly connected to the onboard SATA port on my ASRock Rack MB. Firmware can also be easily extracted out of the update package using 7zip to get the *.bin file to flash it using a different tool or operating system.
Also for HPE drives, support pages refer to them by the HPE model number, "EO000375KWJUC" in the case for the 375GB P4800x, so just try searching for that string on the support website, and there should be a few entries for firmware. (Also the reason I don't prefer using HPE drives, can't tell what model they are by looking at the drive model string, have to look it up to figure out what it is, along with almost non-existent SMART for older SATA drives.)