All the legends when unarmed run around throwing haymakers, push kicks and spinning kicks which I think is a bit bland so I thought it'd be cool if each legend had their own style of unarmed combat.
I'll start by saying that I'm not a proficient martial artist, I did boxe for 3 years when I was younger so my knowledge of martial arts is limited at best, nowadays I'm only a fan or Martial Arts movies and the history behind the fighting forms of the world if anyone with more knowledge stumbles upon this please correct me where I've said wrong or erroneous stuff regarding the fighting forms or think of one that would suit a legend better please correct me or add it.
Loba is of Brazilian descent so I'd say that Capoeira would be the best fighting for for her, also it would fit in with her personality as Capoeira comes from dancing and the classical capoeira is full of flashy moves it would go with her style, her ultimate also reminds me a little of it with the spinning kick not far from a Martelo de Negativa, except instead of using her hands she uses her staff.
Crypto is of Korean descent so Taekwondo would be what I'd imagine him using, plus he runs around with his hands in his pockets so him using mostly his legs would make sense from that point of view.
Bangalore is a former grunt for the IMC, so she'd use something in the military category like Systema or Marine Corps Martial arts, it would also fit with her punisher's style of disarming her opponent and killing them with their own weapon.
Wattson of French descent in would see using Savate with some Capoeira thrown in since she was trained by Loba.
Wraith is a former test pilot, whether she got there through the military division (therefore being along the lines of Bangalore) or the science division of the IMC I don't know if it's ever been made clear lore wise, but she being the tiniest legend I think Krav Maga would suit her as it focuses less on brute strength than most, also her voices from the void would allow her to use it almost perfectly, plus being very aggressive it would suit her and Wraith mains that yeet themselves into combat as soon as they land and either winning the fights or dying right away.
Gibraltar is of Pacific Island descent, I'm not familiar enough with that side of the world to know any particular fighting styles originating from that particular area, if someone knows more perhaps they can give him one but for that I'd say Wrestling as it focuses more on submission and his personality and his reason for joining the games was never killing people, but protecting so I don't think when unarmed he'd default to a severely damaging fighting style.
Bloodhound and it's culture are derived from Northern European and Norse cultures so Glima which is a form of folk wrestling originating from Northern Europe would suit Bloodhound, also in Glima one is to respect their opponent and Bloodhound has that sort of philosophy and mentality where the games are seen as a sporting hunt.
Lifeline is of Caribbean origin and she uses her drumsticks as weapons so I'd say she'd chose to go with something like Stick Licking or Calinda which are stick fighting forms originating from Barbados and Trinidad and Tobago respectively, albeit obscure they are recognised forms, otherwise another one that while not focusing on her "ancestry" still does suit her perfectly would be Eskrima from the Philippines.
Mirage while he has no discernible ancestry he is a bar owner and must have seen his fair share of bar fights since the Frontier is not exactly a chill place and having grown up with four brothers I think the fighting from that would suit him best is Kick Boxing.
Pathfinder should be obvious as he has boxing gloves as his Heirloom and shadow boxes is several of his game intros, also being a robot a punch from him would be utterly devastating.
Caustic seems like the type to want to inflict the most pain to his enemy, armed or not (at least Cuastic pre Wattson thaw in his personality) and Combat Hapkido which focuses on Joint Locks, pressure points and redirecting the opponents force, leverage and position Combat Hapkido also puts a lot of emphasis on trapping (basically grabbing an opponents arms or legs to gain an advantage) which seems perfect for the Toxic Trapper.
Octane is an adrenaline junky speed is his whole mojo, Wing Chun focuses of fast strikes and I think he'd use it if only to see how many hits he could lay on an enemy in 10 seconds using his Stim.
Revenant being an assassin and also a Simulacrum retaining knowledge of several past deaths and mistakes the lead to them, and having had god knows what knowledge uploaded to him every cycle to improve him I would say would use something similar to what John Wick uses, which is a mix of several fighting styles, mostly because John Wick's fighting form focuses on killing as fast and efficiently as possible.