The motion of small probe molecules in a two-dimensional system containing frozen polymer chains was studied by means of Monte Carlo simulations. The model macromolecules were coarse-grained and restricted to vertices of a triangular lattice. The cooperative motion algorithm was used to generate representative configurations of macromolecular systems of different polymer concentrations. The remaining unoccupied lattice sites of the system were filled with small molecules. The structure of the polymer film, especially near the percolation threshold, was determined. The dynamic lattice liquid algorithm was then employed for studies of the dynamics of small objects in the polymer matrix. The influence of chain length and polymer concentration on the mobility and the character of motion of small molecules were studied. Short- and long-time dynamic behaviors of solvent molecules were also described. Conditions of anomalous diffusions' appearance in such systems are discussed. The influence of the structure of the matrix of obstacles on the molecular transport was discussed.