The production of vector mesons in the fragmentation process of polarized quarks is studied within the recursive $\mathrm{string}+{{}^{3}P}_0$ model, improving a previous version of the model in which the production of pseudoscalar mesons only was considered. Two types of couplings of the vector meson to quarks are introduced, their coupling constants being the additional free parameters of the model. The angular distribution of the decay products of the polarized vector meson is deduced from the spin density matrix of the meson and the spin information is propagated along the fragmentation chain taking into account the entanglement of spin states. The new model is implemented in a stand alone Monte Carlo program utilized to investigate in detail kinematic distributions and transverse spin asymmetries. The sensitivity of these observables to the new free parameters is discussed and the Monte Carlo results are compared with experimental data on transverse spin asymmetries.