Will my MOTU Traveler mk3 audio interface work with a Mac mini m4 via Firewire 800 to Thunderbolt adapters?

To have any chance at all of working you will need 3 adapters - Thunderbolt 3 to Thunderbolt 2, the Apple Thunderbolt 2 to Firewire adapter and a FW800 to FW400 cable.

The TB-Firewire adapter is no longer available from Apple but can be found in various used markets typically at a significant premium. There is no guarantee it will work for your MOTU unit especially if MOTU hasn't released an updated & required driver for macOS Sequoia.

There are reports in other threads on these forums that Apple has removed software support for "plug and play" FireWire audio devices from recent versions of macOS.

If your FireWire devices did not require installation of drivers – because they are plug and play – you might not be able to use them any more because of this.

If they did require installation of drivers, those drivers might have been broken by

• The transition from Intel to Apple Silicon. Rosetta 2 can run many Intel applications in translation, but it cannot run Intel-only system code such as Intel-only kernel extensions.
• Changes Apple has been making to deprecate extension mechanisms like kernel extensions (where extension code runs with full system privileges) in favor of new ones that are meant to lock security down tighter.

Catch-22!

I concur with MartinR, perhaps more drastically: unless you find anyone to confirm, chances are low. With midi and audio interfaces, adapters are not your best friends. I had to get rid of a RME FW interface 5 years ago, because some important features just didn’t work with FW to USB adapters. Now imagine with TB….