This is the first time I've tried my hand at homebrewing a class at all so any feedback on balance or anything would be greatly appreciated!
https://docs.google.com/document/d/1aSYGaqGZ3OnLHZfi98XhRC3iikKHTptpok0s3_q Fy2s/edit?usp=drivesdk
Linking the document, because these forums impose a space after 75 characters, breaking the URL and sometimes causing no end of confusion.
So, this is not just a homebrew archetype but also a homebrew Order. Is your intent that the Order is only for a samurai, but not for a cavalier? Because I don't see any reason why it wouldn't work for both. Both the archetype and the Order seem to fit a cavalier more, in terms of flavor.
Archetype
Tools of the Trade - I'm not sure that this ability applies to a boarding pike, because I don't think you can "draw" a pike. A PC is either holding it in their hand, or it's not on their person. Unless there's rules around drawing a weapon I'm not familiar with. Or maybe you can holster spear-like weapons on your back or something; I don't know. (Oh, I just read the description for the weapon, and it mentions how they can be stored on the ship itself in strategic positions. I guess it could apply to drawing it from its place on the ship.)
Order
Keep 'em Busy - I feel like instead of an immediate action, this should allow the PC to make an Attack of Opportunity. This limits how many times a PC can use this ability in one round, and would prevent a strategy of surrounding a target with NPC allies that are likely to miss, and allowing the PC to make up to seven immediate attacks on a single target in one round. That's the most extreme example, but even three or four free attacks on a single target in one round is pretty big. But if these are instead AoOs, and the PC invests in being able to make multiple AoOs in a round, that would be justified.
Ruthless Efficiency - Most abilities that double or expand a crit range in Pathfinder (that I've seen), specifically call out that they don't stack with similar abilities. Consider calling that out for this ability.
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Overall, I think it looks pretty good! If I had more motivation, I'd go and compare it to any other existing sea or ship related archetypes and orders to compare power levels or to check for redundancies.
It looks weak, mostly because the ability which replaces mount isn't much while a mount is a significant ability. The other trades are OK-ish.
BTW, why is corsair spelled with an extra E?