The Large Area Telescope on board the Fermi Gamma-ray Space Telescope has collected the largest ever sample of high-energy cosmic-ray electron and positron events since the beginning of its operation. Potential anisotropies in the arrival directions of cosmic-ray electrons or positrons could be a signature of the presence of nearby sources. We use almost seven years of data with energies above 42 GeV processed with the Pass 8 reconstruction. The present data sample can probe dipole anisotropies down to a level of 10^{-3}. We take into account systematic effects that could mimic true anisotropies at this level. We present a detailed study of the event selection optimization of the cosmic-ray electrons and positrons to be used for anisotropy searches. Since no significant anisotropies have been detected on any angular scale, we present upper limits on the dipole anisotropy. The present constraints are among the strongest to date probing the presence of nearby young and middle-aged sources.