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Lipid vesicles are known to undergo complex conformational transitions, but it remains challenging to systematically characterize nonequilibrium membrane dynamics in flow. Here, we report the direct observation of anisotropic vesicle relaxation from highly deformed shapes using a Stokes trap. Vesicle shape relaxation is described by two distinct characteristic timescales governed by the bending modulus and membrane tension. Interestingly, the fast double-mode timescale is found to depend on vesicle deflation or reduced volume. Experimental results are well described by a viscoelastic model of a deformed membrane. Overall, these results show that vesicle relaxation is governed by an interplay between membrane elastic moduli, surface tension, and vesicle deflation.