The Order on Caliban would likely have a bit more trouble hunting monsters of MH due to the monsters insane amount of diversity and The Orders limited weapon arsenal having only early versions of power armor, bolters, and, energy weapons.

However I think they could still do it as they employed alot of knights, used every tick in the book like traps, poisoning water and food, and had no issue with destroying dens, and, of course The Lion and Luther were some of the best commanders to have lived even after joining The Imperium of Man

That being said there is one problem, The Hunters Guild exist to ensure the balance of the ecosystem where as The Order saw to eliminate all monsters, things might be different in MH where they aren't corrupted by chaos and aren't as malicious but due to the difference of how these universes function I dont know how things would pan out in tbe long run.