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Basically title.
You can't light a fire to cook your food because everything's all wet and there's no air you see. But can you take some raw meat, apply an electric eel to it for 20 minutes, turning once, until the flesh is white and the juices run clear, then serve.
The scientific question is whether the reactions that cook food are purely to do with heat, or do they require oxygen or flame as well. And would electricity underwater produce heat like this?
For starters, electricity and water are not a great combination. Your merfolk are as likely to cook themselves as anything else.
Beyond that, ask yourself if they need to cook at all? The main reasons humans learned to cook is so that we could break down the fibrous cellular structures of roots and grains making them easier to digest, and to preserve meats to make them store longer... But, when you cook, foods become more water soluble which means that their food would just dissolve in the water. Also, aquatic plants and animals don't have the toughness of their terrestrial counter parts; so, your merfolk have way less incentive to need to cook.
This does not mean that your merfolk could not develop a diverse and interesting cuisine though. On the contrary, this is the prefect opportunity to consider how a chefs job could become far more interesting to your audience under different circumstances. You can't make stew because the ingredients float away. You can't sprinkle in some pepper because it would dissolve in the water. Instead, their prepared food will likely involve stuffing, skewering, wrapping, or even weaving foods together.
Sous vide is a popular cooking method that involves putting food in a vacuum bag and sitting it in a warm bath for a few hours. There's no reason merfolk couldn't use the same approach to cook food underwater. Using an electric eel would probably not work because of the energy requirements of heating water. But using a thermal vent or other energy source would make cooking conceptually easy.
Prisoners in USA cook using electricity, by putting electric cables inside pots of water. The water boils in a few seconds, yes black outs are probable.
There's a guy on YouTube who explains how he used to cook pasta in prison this way.
Also electric stoves/heaters/boilers have existed for more than a century now.
And yes, rice cooked in a rice cooker is still considered cooked rice, even if there's no fire.
The answers that mention that people on land cook with boiling water neglect the point, brought up by @OmarL, that water conducts heat very well. When you cook pasta in boiling water on the stove, it works well because the pot is surrounded by air that insulates the pot and allows it to be much hotter than the environment. To cook underwater efficiently, you'd want to build ovens with thick walls of insulating materials. Rock and sand would probably be your best bet. Your goal would be to reduce as much a possible convective transfer of heat between the water in the oven and the ocean outside the oven.
Now, the question is how to heat the ovens. Hydrothermal vents are definitely a good solution @TheSquare-CubeLaw are a good solution. Maybe you could pipe hot water through insulated (mostly rock?) pipes to merfolk houses.
But maybe hydrothermal vents wouldn't be very convenient or nearby merfolk settlements (you have to live where the food sources are). Electrical power could work. We run electrical cables through water all the time, they just need to be properly insulated. But how would merfolk make wires? Now I'm trying to imagine mining of ore, metallurgy, and fabricating insulating materials underwater. Probably some native metals like gold and platinum can be found underwater and used without any processing. Apparently seabed mining ( https://www.nature.com/articles/d41586-019-02242-y ) is now a thing for land-based humans, although I bet much of the processing happens on land.
Most of human technology has been based on burning stuff in air as an energy source and I would guess merfolk would need to do that too. It's the easiest source of concentrated energy for a preindustrial society. So, I imagine that the merfolk would need beach-based or maybe floating facilities where they dry and burn seaweed they collect. They could then operate smelting facilities on rocky outcroppings or on the beach. They could probably build dams and turbines underwater to extract tidal power. Could they build an electric generator that operates underwater? I might guess it would be easier to put it above water or fill a cavity with air for its operation.
The need to operate facilities on beaches or further on land might cause interaction or conflict with land-based creatures.
Maybe they would make floating homes near the surface and just start a fire on a floating platform and place their food below. The oven could be thermally insulated with pumice obtained a nearby volcanic island. Maybe it would just be easier to cook on top of the floating platform?
Maybe they could build electrical transmission circuits by drilling long boreholes through rock in the seafloor and filling these boreholes with seawater. Seawater isn't as good of a conductor as the aluminum wire we used for power transmission, but could be good enough to transmit electricity from the tidal power or ocean current generating station to homes a few hundred meters away.
It's really interesting to think about how technology could develop underwater. Would there be too many engineering challenges, making the merfolk remain in the Stone Age? Or would they be able to find ingenious solutions to doing many things completely underwater? Or would they mostly make use of beach or floating facilities and use technology similar to us?
Cooking requires water and heat. It doesn't matter if the heat comes from a flame or not.
Your merfolk could either place their food close to a thermal vent if they live deep enough. Otherwise, close to the surface, they could use lenses and mirrors to drive sunlight onto a pot above water. A device that works like this is sometimes called a solar oven, but the proper term is solar cooker - see the Wikipedia article for that.
And if your merfolk are able to walk over land like those of Netflix's Disenchantment, then they can make fire just like us humans do.
Yes. Electrical power = current^2 * resistance. We have incandescent lightbulbs underwater, and lightbulbs are just cooking surfaces that produce light as a side-effect. Just be sure to insulate the cooking apparatus from the rest of the ocean lest it dissipate your heat.
Also, there are many materials that burn underwater by providing their own oxygen, which can be used for cooking. Some are products are currently on the market like stormproof matches. You could also use lithium, propane with an oxygen line, etc. Your merfolk can trade for them, or even order them online directly. Note that even if you cook food, it'll get cold quickly underwater. Presumably the merfolk don't mind.
For another option, keep in mind that the cooking means denaturing proteins and killing pathogens. Merfolk can do that with chemicals like lemon juice (https://www.smallfootprintfamily.com/raw-beef-recipes).
The biggest difficulty mermaids run into while cooking is the fact that water conducts heat much better than air. So they find it extremely difficult to keep the heat insulated from their own bodies.
The way to combat this is to make sure that they're standing below the meat that's cooking. This way, the convection brings most of the heat upward, away from your scales. They also need to make sure that a fresh supply of cool water is constantly brought in between the cooking food and themselves, like a thermal cushion.
Incidentally, this is also the reason why mermaids wear seashells on their fronts. Seashells have very low thermal conductivity, so it protects their torsos.
About the simple possibility cook underwatter with electricity - could be used something like this
https://en.wikipedia.org/wiki/Pressure_cooking#Third_generation_%22electric_pressure_cookers%22
I have some at home, it is insulated from the heat inside, so while there is boiling watter inside, hotter then 100C, it is just nice warm outside and can be carried for infinite time in bare hands. I cook a soup with meat there regularry - the meat is totally submerged all the time and definitily cooked better, than on open fire in classic pot.
Out technique would need some small upgrades about electric insulation, which is not problem at all and about heat insulation, which can be solved too.
The problem is, how make electricity and cables under watter. With some handwaving we can make cables from gold and insulation (both electrical and thermal) from some kind of https://en.wikipedia.org/wiki/Natural_rubber (Rubber begins to melt at approximately 180 °C (356 °F). )
If we can get also magnets, we can make https://en.wikipedia.org/wiki/Dynamo with rotating magnets and rubber-gold stationary source of electricity.
But if we want it really mediavial, lets say we make https://en.wikipedia.org/wiki/Diving_suit from some textile, pump there lot of cold watter by textile pipes and use underwatter volcano to harvest lava. The cold watter inside would offset the hot of outside and we get really hot stones by https://en.wikipedia.org/wiki/Pliers which we could put in any properly sized hole with meat, put any https://en.wikipedia.org/wiki/Lid on this improvised pot and we would get cooked meat too :)
(Or be more modern, we can use the volcano to melting metals and make the previous version rubber-gold more precise and even from common metals, if needed)
