Fish breathe through their gills by extracting dissolved oxygen from the surrounding water and most bony fish species have evolved swim bladder to help them maneuver the sea effortlessly. How do the mermaid species thriving on vegetarian diet only lacking any gills and swim bladder live without having to surface for air at all?
The requirements for a herbivorous mermaid with gills or a swim bladder which spends its life underwater are fairly contradictory. To the extent that an organism with all these characteristics would be nonviable. This is a challenge in creature design. They can be considered one by one.
A mermaid with a swim bladder will stay underwater its entire life, because it lacks buoyancy and it will tend to sink. Staying underwater is easy. Indeed it is most likely to spend its life on the bottom. If it's herbivorous and feeds on bottom plants this might not be a problem.
Lacking gills is not a problem. There are marine mammals that come to the surface to breathe. For examples, whales, dolphins and seals. But if the mermaid stays underwater it isn't possible for it to surface and breathe. There is no need for it to have lungs as well. Not so easy to achieve.
Now to consider possible alternative mechanisms for each requirement.
A possible replacement for a swim bladder is a low density body. This will enable the mermaid to float underwater and it is no longer a bottom dweller.
To replace gills and lungs there could be an internal gill system. Not impossible, but tricky. The mermaid would need to "pump" large volumes through its body, through its internal gill cavities to extract sufficient oxygen for respiration. This suggests a mermaid that swims around with its mouth open for the inflow of seawater and for its expulsion from its nether regions. This mermaid would be a living submersible hydrojet.
The reason for is simple. If the mermaid doesn't have gills, and it can't have lungs, then respiration is effectively impossible. The only alternative is an internal gill system.
The energetics involved in a living hydrojet organism are problematic. It would need to consume copious qualities of food to generate the metabolic energy to "pump" the water through its own body and to aerate its internal gills. Herbivores tend to digest their food relatively slowly, this may weight down any mermaid species of this kind. Sinking to the bottom won't help its hydrojet aeration.
There is one further conceivable step for a mermaid of this kind. That would be to abandon the possibility of it being a vertebrate. Bones and vertebrate tissues are relatively heavy biological materials (relative to seawater, that is). If this mermaid was a form of jellyfish that simply happened to have a bodyform that resembled a mythical mermaid then many of the features required for this variety become plausible.
It doesn't need gills (nor lungs too), oxygen can be absorbed from its surrounding medium, or a swim bladder, due to its low density tissues enabling it to float easily in the sea. Jellyfish species don't tend to be herbivorous, but this might not be an impossibility.
One major drawback with mermaids of this kind will be their tendency to collapse into a gelatinous puddle when brought on to land. No romance to be had there.
A mermaid imitation jellyfish may not be what the OP had in mind when he proposed his question, but as squishy and yucky** as it may be, this does fit the bill.
**: Not everybody finds jellyfish "squishy and yucky". They are delightful and beautiful, but quite fragile creatures. Not all Nature's nice creatures are vertebrates.