Plausibility: There is no reason why such a system couldn't work in principle. DWKraus's answer outlines one way, and I outline another way below.
So if it can work in principle, why couldn't you find any examples? One reason might be that there are simpler ways of getting three different phenotypes from a single locus, which might therefore be more likely to evolve in the first place. For example, the alternative mating strategies among male Uta stansburiana lizards—which themselves follow a rock-paper-scissors dynamic—are controlled by a single locus with alleles o, b, and y, where o is dominant to both b and y and y is dominant to b. Hence the orange phenotype is either oo, ob or oy, the yellow phenotype is either yy or yb and the blue phenotype is bb only. (Phenotypically, orange beats blue, blue beats yellow, and yellow beats orange, due to the alternative behaviours exhibited by males of each colour.)
So just to be clear, having three different phenotypes, even if they themselves exist in a rock-paper-scissors nontransitive relationship in terms of competitive fitness, does not require any nontransitive dominance relationship at the genetic level.
Skew: There is no reason why such a genetic architecture would necessarily prevent one phenotype from being more common than the others. Let's say the alleles are called R, P, and S. Suppose that both RR and RS individuals are phenotypically "Rock", PP and PR individuals are phenotypically "Paper" and SS and SP individuals are phenotypically "Scissors". Any combination of "Rock", "Paper", and "Scissors" individuals is possible with such a setup; this is trivial since we can just stipulate that, at a given time, X% of individuals are RR, Y% are PP, and Z% = 100% - X% - Y% are SS. But more generally, if we assume random mating between types, there are two degrees of freedom in the system (the proportion of alleles at the locus that are R and the proportion that are P, with the remaining proportion all being S) and these two degrees of freedom are sufficient to produce any combination of X%, Y%, and Z% = 100% - X% - Y% as outlined above.
The question then is whether any given proportion of the three phenotypes in the population can be maintained stably, particularly in the case where the different phenotypes mate with each other and hence heterozygotes are common. If the proportions are being maintained by negative frequency-dependent selection, such that, for example, the fitness of phenotype-"Rock" individuals is higher than average when there are fewer than some optimal proportion of "Rock" individuals (whether that be 33% or anything else), and the fitness of phenotype-"Rock" individuals is lower than average when there are more than the optimal proportion of "Rock" individuals, this leads to a stable maintenance of that optimum frequency. That's because, provided there is at least one homozygote in the population (in this case RR), the average "Rock" individual is more likely to pass on the R allele than any other allele (since they are either RR or RS), so if the frequency of "Rock"-phenotype individuals is below its optimum, the result is that the transmission of R alleles increases above the average transmission of other alleles. Similarly for P and S, therefore, a stable mix of phenotypes can be maintained.
Note that, in order for an intermediate mix of phenotypes to be stable, you still have to explain what leads to the negative frequency-dependent selection in your population. Or maybe the three types live somewhat separately, so they aren't competing fully for the same resources.
There is a number of ways of getting nontransitive dominance among three possible alleles at a diploid locus, so as to resemble the rock-paper-scissors game. I think the most fun way is just to follow the logic of rock-paper-scissors itself!
In other words, let's call the three alleles R, P, and S. They are translated into proteins Rock, Paper, and Scissors, respectively. Each of these proteins has an active site, which does the actual work of producing the alternative phenotype. Hence, homozygous individuals (RR, PP, or SS) have the phenotype corresponding to the allele they are homozygous for. Each protein also has a secondary site, which plays some role in interaction with other proteins.
In PR heterozygotes, Paper's secondary site binds to Rock's active site. This blocks Rock's active site from doing any work, while leaving Paper's active site exposed on the outside of the Paper-Rock dimer. Paper covers Rock. (If Paper is at least as highly expressed as Rock in heterozygotes, then all Rocks could be covered by a Paper.)
In RS heterozygotes, Rock's secondary site allows it to act as a kind of chaperone causing Scissors to fold up into an inactive, mangled mess that can't unfold itself. Rock is unchanged by this interaction and remains free to do other work via its active site. Rock breaks Scissors.
In SP heterozygotes, Scissors' secondary site cleaves Paper into an inactive form. Scissors is unchanged by this interaction, and remains free to do its own thing. Scissors cuts Paper.
Paper cannot cover Scissors because it only binds to Rock's active site, not Scissors'; and Rock can't break Paper, nor can Scissors cut Rock, because these interactions are specific to the very different shapes of the three proteins.