Yes, I think this is total feasible. From a materials perspective natural rubber (which is derived from latex) is easily up to the aquatic environment (it is often used to make wetsuits). Here are my thoughts on how this would work for various types of clothing.
First, you need to harvest the latex. Many types of plants produce latex, it's actually moderately common. If your world is set on an alternate earth, feel free to read about latex-producing plants and pick the one you think would best suit your needs. Otherwise you can just make up a plant to produce your latex: it's common enough that I don't think anybody would question you.
Now, there are a couple ways you could go to actually make the clothing. One is to make form-fitting clothing, as you mentioned. There are some issues with this, but I think it's workable:
- You don't want the latex to adhere to the body once it has formed into a rubber. This is easily solved with some kind of body lubrication.
- You need to allow the latex to dry without rubbing it off. This would is a little tricky. For a human, I would say they could stand there until the layer is cured, and then re-dip their feet to fix the part that was messed up by standing. Something similar might be possible with mermaids if they can stand on their tails well, otherwise you need another solution. Perhaps solving this suggest a different application method (perhaps it is painted on in sections, instead of dipping the whole body at once).
- As mentioned by Kaiannae, the suit will be hard to get back on once the wearer is cut out of it for the first time. This could be ameliorated by cutting the clothing into sections and then binding them back together less tightly, but this eliminates the main advantage of this type of form-fitting clothing (namely, the ability to hold a water layer near the body, providing insulation from cold waters). It may also be solvable using the same body lubrication as from (1).
- This is less of a technical problem, but one of the main purposes of clothes at least in human societies is to outwardly display something about the wearer, be that their individuality, a group identity (i.e. uniforms), or some personal trait. Clothing made in this way would have less potential for fashion or decoration than a more "traditional" type of clothing. This may not be a problem, depending on the society of your mermaids (maybe they're very collectivist, and the uniformity helps them form a strong group identity), but it's worth thinking about.
Now, there is another possibility. Latex could be cast in flat, thin layers and then bound together into garments. These garments would do little for keeping warm, but they have some other advantages over form-fitting clothing:
- They are much easier to make. All you need to do is make a flat surface - maybe a stone that's been ground flat - and pour the latex onto it, maybe with a layer of lubricant to prevent adhesion, and maybe with a tool to smooth the layer to be thin. Then you let it cure, and cut the sheets into strips or other shapes to form the elements of clothing. Bindings could be made from thin strips of this same material, if a different material doesn't suit better.
- You could make many different styles of clothing: skirts made from strips or sheets, shirts with many layers to act as armor, fashion accessories like armbands or tassels, and so on.
- You can modify the latex itself easily during casting. For example, mixing in plant fibers would yield a tougher, less stretchy piece of rubber. Mixing in sand or some other powdered ceramic may strengthen the material for armor, or could be used to add colors. If they have advanced enough technology, they could vulcanize the rubber to make particularly strong and rigid pieces. Some of these things could be done with a body-dip or paint-on method, but not all, and it may be easier in flat castings.
Overall, I think there's no reason to restrict yourself to one way or the other. Maybe certain things are made to be personalized and form-fitting (gloves, wet suits for keeping warm, tail guards, etc) and others (skirts, armor, decorative clothing) are made the other way. Having multiple methods adds some believability to your world: it's rare for there to only be one way that something is done.