Perform a montage with the frogmen showing how species doesn't matter what sort of person you are. Live in a utopian happiness with the frogmen, merging into the herd and supporting their community.
Once you have their trust, lure them into smoking assorted herbs. Once they form a dependency, restrict supply and become extortionate, demanding more and more services from them for the herbs. There will doubtless be some more outspoken and troublesome of the frogmen - they must be dealt with with discretion.
Even after the immediate labour needs of the tribe are fulfilled, manipulate the frogmen to build various megaprojects. The point of this is not to benefit from the megaprojects, but rather to change the social structure of the frogmen. The once-mighty creatures, cripples by their addiction for the assorted herbs, shall live in a broken society. Living conditions for the frogmen must be reduced so that they are only just surviving: they will not be aware of the long-term destruction of their society.
With time, the place of frogmen will be firmly established. Broken creatures, they will beg and toil for the elves just for the next sweet hit of the assorted herbs. Elves will be established as the superior species, and as long as the living conditions of the frogmen is kept poor, they will be unable to uprise and rebel. Of course, in order to maintain this, the elves need to establish a strict society: even though elves will be equal to frogmen by the letter of the law, the practice will be different. By exaggerating the goblin menace as the root of all evils making scapegoats of troublesome frogmen (claiming them to be agents for the goblins), a clear enemy can be established for the frogmen to concern themselves with.
By offering large rewards of assorted herbs for any potential goblin agent frogmen, any frogman that tries to disrupt proceedings will be quickly turned in. They will also be unable to identify with any other sympathisers, as they will not be able to tell them from the regular members of society.
A secret police will have to be established. The frogmen should be constantly living in fear, unable to shoe a moment's deviance from acceptable thoughts. The actual logistics of monitoring every frogman will be a huge problem, so the frogmen must be unable to know if they are being watched or not. By establishing a series of underground tunnels underneath the frogman settlements where elves and frogmen can be posted, the frogmen will be able to be watched and heard at any time, but they will not know at any given point whether they are being monitored or not. They must therefore always act like they are being monitored, in case they are.
The knowledge and society of the elves must be heavily censored, too. The elves must establish a leader for the frogmen to follow. The leader must himself be an frogman, called "amazing amphibian" (or alternatively, the turtle). Pictures of this leader will be placed everywhere. Since the leader is an frogman, the other frogmen will be encouraged to act like him, and with time they will want to act like him. All new reaching the frogmen must be completely fabricated: they must be told how they live in an age of prosperity thanks to the glory of their leader and his following of the desired way of life, despite the filthy goblins who shall be seen as all that is bad. This leader does not need to actually exist; the frogmen need to think that he exists.
Over time, a stable society will be formed. Frogmen will be brainwashed into pure hatred of the goblins, and loving their leader and his agents, the elves. They shall see the leader and the elves as the source of all that is good: Their society, their defence against goblins and their assorted herbs. The mere concept of society as anything different will be outside what they can conceive: the language that they use shall have to be changed after a time to make even expressing an unfavourable thought impossible. They shall live in constant fear, though. They shall be terrified of goblin agents, and that they might be thought to be goblin agents.
Disappearances in such a society would be common, and thus nobody would notice when a frogman is taken away to be traded to the dwarves. They will not be told of the dwarves, though. Whenever a frogman is needed for trading, one shall be taken, a problematic one if preferable.