Observation of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>Y</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>4230</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> and evidence for a new vector charmoniumlike state <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>Y</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>4710</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>e</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>e</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mo stretchy="false">→</mml:mo><mml:msubsup><mml:mi>K</mml:mi><mml:mi>S</mml:mi><mml:mn>0</mml:mn></mml:msubsup><mml:msubsup><mml:mi>K</mml:mi><mml:mi>S</mml:mi><mml:mn>0</mml:mn></mml:msubsup><mml:mi>J</mml:mi><mml:mo>/</mml:mo><mml:mi>ψ</mml:mi></mml:math>

Cross sections for the process ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}_{S}^{0}{K}_{S}^{0}J/\ensuremath{\psi}$ at center-of-mass energies from 4.128 to 4.950 GeV are measured using data samples with a total integrated luminosity of $21.2\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ collected by the BESIII detector operating at the BEPCII storage ring. The $Y(4230)$ state is observed in the energy dependence of the ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}_{S}^{0}{K}_{S}^{0}J/\ensuremath{\psi}$ cross section for the first time with a statistical significance of $26.0\ensuremath{\sigma}$. In addition, an enhancement around 4.710 GeV, labeled as the $Y(4710)$, is seen with a statistical significance of $4.2\ensuremath{\sigma}$. There is no clear structure around 4.484 GeV. Using a fit with a coherent sum of three Breit-Wigner functions, we determine the mass and width of the $Y(4230)$ state to be $4226.9\ifmmode\pm\else\textpm\fi{}6.6\ifmmode\pm\else\textpm\fi{}22.0\text{ }\text{ }\mathrm{MeV}/{c}^{2}$ and $71.7\ifmmode\pm\else\textpm\fi{}16.2\ifmmode\pm\else\textpm\fi{}32.8\text{ }\text{ }\mathrm{MeV}$, respectively, and the mass and width of the $Y(4710)$ state to be $4704.0\ifmmode\pm\else\textpm\fi{}52.3\ifmmode\pm\else\textpm\fi{}69.5\text{ }\text{ }\mathrm{MeV}/{c}^{2}$ and $183.2\ifmmode\pm\else\textpm\fi{}114.0\ifmmode\pm\else\textpm\fi{}96.1\text{ }\text{ }\mathrm{MeV}$, respectively, where the first uncertainties are statistical and the second are systematic. In addition, the average Born cross section ratio $\frac{{\ensuremath{\sigma}}^{\text{Born}}({e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}_{S}^{0}{K}_{S}^{0}J/\ensuremath{\psi})}{{\ensuremath{\sigma}}^{\text{Born}}({e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}J/\ensuremath{\psi})}$ is measured to be ${0.388}_{\ensuremath{-}0.028}^{+0.035}\ifmmode\pm\else\textpm\fi{}0.016$, or ${0.426}_{\ensuremath{-}0.031}^{+0.038}\ifmmode\pm\else\textpm\fi{}0.018$ if three-body phase space is considered.