Kevlar bones seem improbable. "Kevlar is synthesized in solution from the monomers 1,4-phenylene-diamine (para-phenylenediamine) and terephthaloyl chloride in a condensation reaction yielding hydrochloric acid as a byproduct. The result has liquid-crystalline behavior, and mechanical drawing orients the polymer chains in the fiber's direction. Hexamethylphosphoramide (HMPA) was the solvent initially used for the polymerization, but for safety reasons, DuPont replaced it by a solution of N-methyl-pyrrolidone and calcium chloride."
Lifeforms with a biochemistry similar those found in Earth's biosphere would have difficulty accommodating the hydrochloric acid by-product from its synthesis. Possibly organisms with an exotic biochemistry might be able to handle it safely.
One of the main problems for any organism trying to grow kevlar is that it would have to consume large quantities of the monomers 1,4-phenylene-diamine (para-phenylenediamine) and terephthaloyl chloride needed for the polymerization to form kevlar. In our biosphere organisms only to access sufficient quantities of calcium compounds to make bones.
If there was an environment where the base compounds were found in abundance, then it is conceivable that lifeforms capable of growing kevlar might develop and exist. They would be alien organisms with very exotic biochemistries. Unfortunately, this seems extremely unlikely.