Aerosols in the atmospheres of cloudy gas giant exoplanets and brown dwarfs scatter and polarize these objects' thermal emission. If such an object has an oblate shape or nonuniform cloud distribution, the net degree of linear polarization can show an increase ranging from several tenths of a percent to a few percent. Modern high-contrast imaging polarimeters are now poised to detect such low-polarization signals, opening up a new window into the rotational velocities and cloud properties of substellar companions to nearby stars. In this paper, we present the results of a near-IR survey searching for linearly polarized thermal emission from a sample of two planetary-mass companions and five brown dwarf companions using GPI and SPHERE-IRDIS. We probe the subpercent linear polarization regime that typifies polarized free-floating brown dwarfs and place limits on each object's degree of linear polarization. We relate our upper limits on each target's degree of linear polarization to its rotation rate, and place our results in the context of rotation rates measured using high-resolution spectroscopy.