  We focus on two repeating fast radio bursts (FRBs) recently detected by the
CHIME/FRB experiment in 2018--2019 (Source 1: 180916.J0158+65, and Source 2:
181030.J1054+73). These sources have low excess dispersion measures (DMs) ($ <
100 \rm \, pc \, cm^{-3}$ and $ < 20 \rm \, pc \, cm^{-3}$, respectively),
implying relatively small maximal distances. They were repeatedly observed by
AGILE in the MeV--GeV energy range. We do not detect prompt emission
simultaneously with these repeating events. This search is particularly
significant for the submillisecond and millisecond integrations obtainable by
AGILE. The sources are constrained to emit a MeV-fluence in the millisecond
range below $F'_{MeV} = 10^{-8} \, \rm erg \, cm^{-2}$ corresponding to an
isotropic energy near $E_{MeV,UL} \simeq 2 \times 10^{46}\,$erg for a distance
of 150 Mpc (applicable to Source 1). We also searched for $\gamma$-ray emission
for time intervals up to 100 days, obtaining 3$\,\sigma$ upper limits (ULs) for
the average isotropic luminosity above 50 MeV, $L_{\gamma,UL} \simeq
\,$(5-10)$\,\times 10^{43} \rm \, erg \, s^{-1}$. For a source distance near
100 kpc (possibly applicable to Source 2), our ULs imply
$E_{MeV,UL}\simeq10^{40} \rm erg$, and $L_{\gamma,UL} \simeq \,$2$\,\times
10^{37} \rm \, erg \, s^{-1}$. Our results are significant in constraining the
high-energy emission of underlying sources such as magnetars, or other
phenomena related to extragalactic compact objects, and show the prompt
emission to be lower than the peak of the 2004 magnetar outburst of SGR 1806-20
for source distances less than about 100 Mpc.
