The low energy excited $0_{2,3}^+$ states in $^{96}$Sr are amongst the most prominent examples of shape coexistence across the nuclear landscape. In this work, the neutron $[2s_{1/2}]^2$ content of the $0_{1,2,3}^+$ states in $^{96}$Sr was determined by means of the d($^{95}$Sr,p) transfer reaction at the TRIUMF-ISAC2 facility using the SHARC and TIGRESS arrays. Spectroscopic factors of 0.19(3) and 0.22(3) were extracted for the $^{96}$Sr ground and 1229~keV $0^+$ states, respectively, by fitting the experimental angular distributions to DWBA reaction model calculations. A detailed analysis of the $gamma$-decay of the isomeric $0_3^+$ state was used to determine a spectroscopic factor of 0.33(13). The experimental results are compared to shell model calculations, which predict negligible spectroscopic strength for the excited $0^+$ states in $^{96}$Sr. The strengths of the excited $0_{2,3}^+$ states were also analyzed within a two-level mixing model and are consistent with a mixing strength of $a^2$=0.40(14) and a difference in intrinsic deformations of $|Delta beta|=0.31(3)$. These results suggest coexistence of three different configurations in $^{96}$Sr and strong shape mixing of the two excited $0^+$ states.
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