What materials in the environment could merpeople use to make armor? The merpeople live in the mediterranean sea and have contact and trade with humans who could provide resources and craft the armor for them if necessary. The humans are around classical antiquity in terms of technology.
I'd perhaps look to inspiration from real life, specifically the ancient world. Armor as we often think of it, medieval plate armor, really only existed for a brief period of time. Most of history however, armor was made by layering different materials together, often things like fibers.
This video from How to make everything goes over some stuff about the practice. You might look at what underwater plants might work.
Maybe shells? They could be drilled and linked together, maybe into something resembling scale-mail. There is a large variety of different kinds of shells, which could lead to different armour types.
I think a problem with aquatic armour would be the degradation of the materials of time - a lot of things break down easier in water than while dry. Certain tough aquatic plants may work as a binding fibre, or they could be replaced every now and again.
Bone and teeth could also be used, especially to adorn the amour or make it harder to attack. Some seaweeds are pretty strong, too.
Chitin would be abundant
Merfolk would have learned to farm coral and kelp, and also be skilled in "crustacean husbandry" and have pastures of crabs and similar aquatic animals. Chitin can be fairly tough and can be used or combined together to make composite armor plates. Especially combined with hides of sharks and adorned with fish-scales and pearls. I'd imagine they'd grow giant shells in "moulds" that would eventually become pieces of armor. They'd likely have far better segmented armor than their contemporaries, since they could study the movement of shrimps and crayfish in action.
Other than that there would probably be hydrothermal vents along the tectonic fault lines that could produce native elements and basalt and the temperatures necessary to shape them. Again, connecting a "mould" to a vent would presumably create a shaped fit of a piece of armor over time.
The sea is far less explored than the surface, likely the imagination is less spectacular than reality as for the opportunities of the great below.
Tanned hides of big merpeople.
These hides are tanned so that they are tough and resilient, and slits made to articulate at the joints. They are then put on as armor by small merpeople. This way they still look like merpeople but large and unexpressive ones with skin that is not shiny. Armored merpeople seem impervious to harm. Their enemies suspect they are zombies.
The large live merpeople cannot fit into this kind of armor, but you do not generally hear them complaining too loudly about that to the small merpeople.
Whatever the armour is made of, it should not be too heavy or constrain the movements. Preference should go to fast swimming and agility, since in water it is easier to dodge a blow that should be the primary defence.
The idea of Stian Yttervik of using shark hides on top is good because they reduce resistance to the water flow. Underneath I would use one or more layers of a a textile made of Byssus.
For the heads wooden helmets would be the best solution. Humans should sell a substance that is a mix of lacquer and resins, that would be used to impregnate the wood and make it more resistant to rot and a little bit heavier in order to have neutral buoyancy.
While metal wouldn't be available to merfolk alone, if there are land dwelling species then there will be lots of metal flowing into the merfolk's lands
The armours can't be purely metal, though: They will need some sort of waterproof covering to avoid rust or other damage, but this should be achieveable, regardless of the technological level