Using 4.479×10^{8} ψ(3686) events collected with the BESIII detector, we search for the decays ψ(3686)→e^{+}e^{-}χ_{cJ} and χ_{cJ}→e^{+}e^{-}J/ψ, where J=0, 1, 2. The decays ψ(3686)→e^{+}e^{-}χ_{cJ} and χ_{cJ}→e^{+}e^{-}J/ψ are observed for the first time. The measured branching fractions are B(ψ(3686)→e^{+}e^{-}χ_{cJ})=(11.7±2.5±1.0)×10^{-4}, (8.6±0.3±0.6)×10^{-4}, (6.9±0.5±0.6)×10^{-4} for J=0, 1, 2, and B(χ_{cJ}→e^{+}e^{-}J/ψ)=(1.51±0.30±0.13)×10^{-4}, (3.73±0.09±0.25)×10^{-3}, (2.48±0.08±0.16)×10^{-3} for J=0, 1, 2, respectively. The ratios of the branching fractions B(ψ(3686)→e^{+}e^{-}χ_{cJ})/B(ψ(3686)→γχ_{cJ}) and B(χ_{cJ}→e^{+}e^{-}J/ψ)/B(χ_{cJ}→γJ/ψ) are also reported. Also, the α values of helicity angular distributions of the e^{+}e^{-} pair are determined for ψ(3686)→e^{+}e^{-}χ_{c1,2} and χ_{c1,2}→e^{+}e^{-}J/ψ.