This paper presents a distributed finite-time adaptive integral-sliding-mode (ISM) control approach for a platoon of vehicles consisting of a leader and multiple followers subjected to bounded unknown disturbances. In order to avoid collisions among the vehicles, control protocols have to be designed to ensure string stability of the whole vehicle platoon. First, the constant time headway (CTH) policy known to improve string stability is applied to the case of zero initial spacing errors. Contrary to requiring zero initial spacing and zero initial velocity errors simultaneously in existing methods based on constant spacing (CS) policy, initial velocity errors here are not required to be zero. Then, since string stability condition can fail at the initial conditions, a modified CTH policy is constructed to overcome string instability caused by nonzero initial spacing errors. Moreover, the proposed adaptive ISM control schemes can be implemented without the requirement that the bounds of the disturbances be known in advance. In addition, one effective method is proposed to reduce the chattering phenomenon caused by the indicator function. Finally, simulation results are included to demonstrate its effectiveness and advantages over existing methods.