We present the results of a single dish survey toward 95 VeLLOs in optically thick (HCN 1-0) and thin (N2H+ 1-0) lines performed for the purpose of understanding the physical processes of inward motions in the envelopes of the VeLLOs and characterizing their true nature. The normalized velocity differences (δVHCN ) between the peak velocities of the two lines were derived for 41 VeLLOs detected in both lines. The δV distribution of these VeLLOs is found to be significantly skewed to the blue, indicating the dominance of infalling motions in their envelopes. The infall speeds were derived for 15 infall candidates by using the HILL5 radiative transfer model. The speeds were in the range of 0.03 km/s to 0.3 km/s, with a median value of 0.16 km/s, being consistent with the gravitational free-fall speeds from pressure-free envelopes. The mass infall rates calculated from the infall speeds are mostly of the order of 10^-6 M⊙/yr with a median value of 3.4 ± 1.5 × 10^-6 M⊙/yr. These are found to be also consistent with the values predicted with the inside-out collapse model and show a fairly good correlation with the internal luminosities of the VeLLOs. This again indicates that the infall motions observed toward the VeLLOs are likely to be due to the gravitational infall motions in their envelopes. Our study suggests that most of the VeLLOs are potentially faint protostars while two of the VeLLOs could possibly be proto-brown dwarf candidates.