We present small-angle neutron scattering(SANS) studies of the superconducting vortices in Ba(Fe_0.93Co_0.07)₂As₂. A highly disordered vortex configuration is observed up to the highest measured field of 3T. The field dependence of the magnitude of the SANS scattering vector indicates small vortex lattice domains of rhombic symmetry or a vortex glass with short-range hexagonal order. The disordering is attributed to strong single-vortex pinning. This is supported by an analysis of the radial width of the intensity distribution, which indicates a correlation length of a few vortex spacings. Measurements of the scattered intensity are found to decrease with increasing magnetic field, consistent with theoretical models and in reasonable agreement with estimates of the upper critical field. The temperature dependence is found to follow the BCS s-wave prediction.