FRB 180916 is a most intriguing source at 150 Mpc distance capable of producing repeating fast radio bursts with a periodic 16.35 day temporal pattern. We report on the X-ray and $γ$-ray observations of FRB 180916 obtained by AGILE and Swift. We focused on the recurrent 5-day time intervals of active radio bursting and present results obtained on Feb. 3 - 8; Feb. 25; Mar. 5 - 10; Mar. 22 - 28, 2020 during a multiwavelength campaign involving high-energy and radio observations. We also searched for temporal coincidences at millisecond timescales between all known radio bursts of FRB 180916 and X-ray and MeV events detectable by AGILE. We do not detect any simultaneous event or any extended X-ray and $γ$-ray emission on timescales of hours/days/weeks. Our cumulative X-ray (0.3-10 keV) flux upper limit of $5 \times\,10^{-14} \rm \, erg \, cm^{-2} s^{-1}$ (obtained during 5-day active intervals) translates into an isotropic luminosity upper limit of $L_{X,UL} ∼ 1.5 \times\, 10^{41} \rm erg \, s^{-1}$. Observations above 100 MeV over a many-year timescale provide an average luminosity upper limit one order of magnitude larger. These results provide the so-far most stringent limits on high-energy emission from FRB 180916 and constrain the dissipation of magnetic energy from a magnetar-like source of radius $R_m$, internal magnetic field $B_m$ and dissipation timescale $τ_d$ to satisfy the relation $R_{m,6}^3 B_{m,16}^2 τ_{d,8}^{-1} \lesssim 1$, where $R_{m,6}$ is $R_m$ in units of $10^6$ cm, $B_{m,16}$ is $B_m$ in units of $10^{16}$ G, and $τ_{d,8}$ in units of $10^8$ s.