Do unsaturated fatty acids with an odd number of carbons exist?
C21:3 is a stable fatty acid that could indicate a variety of double-bond isomers.  It is "unusual" in biology because of the odd number of carbons, but that doesn't preclude its stability and accessibility to organic synthesis.  If someone really wanted to make this acid, they could do so easily.
Such a compound, 5Z,14Z,17Z-heneicosatrienoic acid,  has been reported in the natural environment.  It was found in a deep-sea clam, Calyptogena phaseoliformis, that lives in or near hydrothermal vents and lives off of the organic matter produced by chemosynthetic bacteria that live in close proximity to it.  So in addition to being synthetically accessible, it is also made in living organisms.


Thus, for all reasonable interpretations of "exist" in your question, does this fatty acid even exist?, the answer is yes.