The detection of a stochastic signal by recent pulsar timing array (PTA) collaborations, including NANOGrav, PPTA, EPTA+InPTA, CPTA, and MPTA, has opened a new window to explore gravitational waves (GWs) at nanohertz frequencies. Motivated by the possibility that such a signal could arise from primordial gravitational waves (PGWs), we investigate the implications of tensor non-Gaussianity for the PGW power spectrum. Utilizing PTA datasets, we provide constraints on the local-type tensor non-Gaussianity parameter ℱNL. We find |ℱNL| ≲7.97 for a log-normal PGW power spectrum. Our analysis reveals that even moderate tensor non-Gaussianity can lead to significant deviations from standard predictions, thereby offering a novel means to test inflationary scenarios and probe the underlying dynamics of the early Universe. Future multiband GW observatories, such as LISA, Taiji, and TianQin, will be instrumental in complementing these efforts and further refining our understanding of tensor non-Gaussianity.