Assuming a character has eyes set into their palms rather than their face (not intended to be practical or a good thing), how drastically would this impede combat ability, and what (if anything) could be done about it?
If their eyes were set in their palms, rather than their face then they would use their hands to see rather than fight. This would limit their attacking considerably, because they also would not be able to, really, make use their arms. Thus, their hands become the part of the body they are most focused on protecting.
They’ll use their legs and their feet, possibly their body, and almost certainly their head. They may also bite more.
The vast majority of fighting styles you’re looking at are based around having your eyes and other important organs within a human body because they’re designed for people. If this person is basically human and just has their eyes in their hands (and nowhere else), then they are going to be remarkably fragile. The hands are incredibly easy to lock up and segregate off from the rest of the body. If they close their fingers into fists to protect them, then they blind themselves.
A basic rule of thumb to understand about your body is this: the further away it is from your head and your center then the less you need it. Human evolution has essentially structured us so that everything we need to survive is collected in a fairly easy to protect place. Your limbs are what you use to protect that center. (When I say “need”, I mean “absolutely 100% necessary for your body to continue functioning in the worst case scenario”, this is your heart and your brain.) If you move a body part such as an eye to somewhere else on your body, then you are not only changing the dynamic of how it functions but also it’s level of importance.
With their eyes moved to their hands, (assuming you want them to continue being able to do similar every day activities such as cooking, cleaning, picking up objects, and generally caring for themselves) then, since they’ve been moved to a much more risky spot, they have dropped significantly on the scale of necessary body parts and their vision will also be much more limited.
When coming up with rules for this character, you may want to consider the sight being a non-essential, or less essential sense when compared to the other four. Touch is also a little screwy due to the eyes being in the hands. They may simply have a much better sense of smell or hearing to compensate for the lack of sight.
It’s a matter of adaptation and this is where looking into different fighting styles for an idea of how they might fight won’t help you. All combat styles in the real world are developed for humans. They are designed with the idea that your eyes are going to be in your head and that you (as a non-impaired human) will be highly dependent on them. If you are set on using real world fighting styles, then you may want to look into teaching tactics for blind students. There are quite a few examples out there of the blind practicing martial arts. Again, though, this character is going to have different concerns.
What you should probably start doing, if you haven’t already, is to begin tracking what you use your hands for everyday. When you are performing everyday activities start imagining what it would be like if your eyeballs were in your palms. If you are in class, how would you read a whiteboard? How would you take notes? How do you hold a hot cup of coffee?
Begin coming up with strategies the character might use to adapt in similar situations. Beyond combat, this is what’s going to be key to writing the character. It’s a neat idea to play with, but you’re going to have to be the one to figure out how to make it work on the page. Figuring out how the character goes about their everyday, what they use their limbs for, how they’ve adapted, will be key toward figuring out how a character fights.
On the most basic of basic levels martial combat is understanding its structured around the basic idea that certain body parts need to be protected. The same ones to be protected are also the points one attacks in an opponent. If you haven’t discovered what parts of their body this character needs to protect, then figuring out how they fight is going to be very difficult.
Most fighting styles make substantial use of the palm in defense. While one can use the forearm and wrist, grabbing is necessary for grappling, and using the hand can provide more control over the opponent when it comes to deflection and/or transitioning into holds. When attacking with the palm, the heel is the primary method of striking, but open hand strikes such as slaps and boxing the ears are common. Slaps are also very common in instances where someone wishes to prolong the combat period and leave fewer signs of a struggle due to the force of the blow being spread over a wider part of the body. This is the primary reason why many abusers slap instead of punch, and its where the term “slapping them around” comes from.
The lack of hand usage is a substantial blow to their combat ability. However, what is even greater blow is the part where the hands are in constant motion. For reference, it would be like watching the world with a particularly rapid form of fast cuts and shaky cam. Assuming their eyes work how human eyes normally function then they would constantly be losing sight of their enemy. If they are sight reliant, then that is very bad. They also, depending on how their eyes function, may not be able to judge distance well.
The average/non-impaired fighter will only really lose sight of their opponent when they spin and even then, it’s only for a fraction of a second. This assumes that neither sweat nor blood impede their vision. This character may be forced to close their fingers all the time. If they don’t, they’ll still need to turn their hands back toward themselves in order to protect the inside of their arms (where the veins are). I know my hands become very sweaty when I get nervous/work out, so that may be another concern. Their eyes are also in danger if they fall. When we fall, we often try to use our hands to catch ourselves (which can lead to injury anyway). This is obviously a problem for this character and it’s going to hurt their utility if they have no adaptations to support it.
Again, though, if their eyes aren’t the main sense they use to perceive the world around them then the discussion changes direction entirely. I suggest looking into insects and crustaceans that have their eyes separated/unprotected from their body to get ideas.
This was just stuff I thought of. Hopefully, you’ll be able to find more.