Discovering new actively accreting protoplanets is crucial to answering open questions about planet formation. However, identifying such planets at orbital distances where they are expected to be abundant is extremely challenging, both due to the technical requirements and large distances to star-forming regions. Here we use the kernel phase interferometry (KPI) technique to search for companions around the ∼6 and ∼8 Myr old Herbig Ae stars MWC 758 and MWC 480. KPI is a data analysis technique that is sensitive to moderate asymmetries, arising from, e.g., a circumstellar disk or companions with contrasts of up to 6–8 mag, at separations down to and even below the classical Rayleigh diffraction limit (∼1.2λ/D). Using the high-spectral-resolution K-band mode of the SCExAO/CHARIS integral field spectrograph, we search for both excess Br-γ line emission and continuum emission from companions around MWC 480 and MWC 758. We are able to set limits on the presence of rapidly accreting protoplanets and brown dwarfs between 4 and 16 au, well interior to those of previous studies. In Br-γ, we set limits on excess line emission equivalent to accretion rates ranging from
10
−
5
M
j
2
.
yr
−
1
to
10
−
6
M
j
2
.
yr
−
1
. Our achievable contrasts demonstrate that KPI using SCExAO/CHARIS is a promising technique to search for giant accreting protoplanets at smaller separations compared to conventional imaging.