The dipole polarizabilities, dipole hyperpolarizabilities, quadrupole moments, and quadrupole polarizabilities of the $5{s}^{2}\phantom{\rule{0.16em}{0ex}}^{1}S$ and $5s5p\phantom{\rule{0.16em}{0ex}}{}^{3}{P}^{o}$ states of ${\mathrm{In}}^{+}$ and Sr are calculated by using the finite-field method. The electron correlation effect and the basis set convergence are investigated in the relativistic coupled-cluster and configuration interaction calculations in order to obtain polarizabilities of high accuracy. Comparative study of the fully and scalar relativistic calculations reveals the effect of the spin-orbit coupling on the dipole polarizabilities of ${\mathrm{In}}^{+}$ and Sr. The blackbody-radiation shifts of the clock transition $5{s}^{2}\phantom{\rule{0.16em}{0ex}}{}^{1}{S}_{0}\ensuremath{-}5s5fp\phantom{\rule{0.16em}{0ex}}{}^{3}{P}_{0}^{o}$ due to the dipole polarizabilities, quadrupole polarizabilities, and dipole hyperpolarizabilities are evaluated to be 0.017, $8.33\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}$, and $1.93\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}17}$ Hz for ${\mathrm{In}}^{+}$ and 2.09 and $5.82\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}$, and $1.69\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}15}$ Hz for Sr. The blackbody-radiation shifts from the quadrupole polarizabilities and dipole hyperpolarizabilities are significantly less than that from the dipole polarizabilities and therefore can be safely omitted for the quoted ${10}^{\ensuremath{-}18}$ uncertainty of the optical frequency standard of ${\mathrm{In}}^{+}$ and Sr.