Active galactic nuclei (AGNs) with intermediate radio powers are capable of driving multiphase outflows in galaxy bulges, and are also more common than their high-radio-power counterparts. In-depth characterization of the typical host galaxies and likely triggering mechanisms for this population is therefore required in order to better understand the role of radio AGN feedback in galaxy evolution. Here, we use deep optical imaging data to study the detailed host morphologies of a complete sample of 30 local radio AGNs with high-excitation optical emission (HERG) spectra and intermediate radio powers [ z < 0.1; 22.5 < log(L_1.4GHz) < 24.0 W Hz-1]. The fraction of hosts with morphological signatures of mergers and interactions is greatly reduced compared to the 2Jy radio-powerful galaxies [log(L_1.4GHz) > 25.0 W Hz-1] with strong optical emission lines: 53 ± 9 per cent compared with 94 ± 4 per cent. In addition, the most radio-powerful half of the sample has a higher frequency of morphological disturbance than the least radio-powerful half (67 ± 12 per cent and 40 ± 13 per cent, respectively), including the eight most highly disturbed galaxies. This suggests that the importance of triggering nuclear activity in high-excitation radio galaxies (HERGs) through mergers and interactions reduces with radio power. Both visual inspection and detailed light profile modelling reveal a mixed population of early-type and late-type morphologies, contrary to the massive elliptical galaxy hosts of radio-powerful AGNs. The prevalence of late-type hosts could suggest that triggering via secular, disc-based processes has increased importance for HERGs with lower radio powers (e.g. disc instabilities and large-scale bars).