In the presence of P, T-violating interactions, the exchange of axionlike particles between electrons and nucleons in atoms and molecules induces electric dipole moments (EDMs) of atoms and molecules. We perform calculations of such axion-exchange-induced atomic EDMs using the relativistic Hartree-Fock-Dirac method including electron core polarization corrections. We present analytical estimates to explain the dependence of these induced atomic EDMs on the axion mass and atomic parameters. From the experimental bounds on the EDMs of atoms and molecules, including ^{133}Cs, ^{205}Tl, ^{129}Xe, ^{199}Hg, ^{171}Yb^{19}F, ^{180}Hf^{19}F^{+}, and ^{232}Th^{16}O, we constrain the P, T-violating scalar-pseudoscalar nucleon-electron and electron-electron interactions mediated by a generic axionlike particle of arbitrary mass. Our limits improve on existing laboratory bounds from other experiments by many orders of magnitude for m_{a}≳10^{-2} eV. We also place constraints on CP violation in certain types of relaxion models.