We present measurements of second-order azimuthal anisotropy (v2) at midrapidity (|y|<1.0) for light nuclei d,t,He3 (for sNN=200, 62.4, 39, 27, 19.6, 11.5, and 7.7 GeV) and antinuclei d; (sNN=200, 62.4, 39, 27, and 19.6 GeV) and Hē3 (sNN=200 GeV) in the STAR (Solenoidal Tracker at RHIC) experiment. The v2 for these light nuclei produced in heavy-ion collisions is compared with those for p and p. We observe mass ordering in nuclei v2(pT) at low transverse momenta (pT<2.0 GeV/c). We also find a centrality dependence of v2 for d and d. The magnitude of v2 for t and He3 agree within statistical errors. Light-nuclei v2 are compared with predictions from a blast-wave model. Atomic mass number (A) scaling of light-nuclei v2(pT) seems to hold for pT/A<1.5GeV/c. Results on light-nuclei v2 from a transport-plus-coalescence model are consistent with the experimental measurements.