Can doing kegel exercises get you sore on the inside of your lower stomach?

Shouldn't. Kegel exercises done correctly are supposed to increase the strength of the elevator plate. That is a group of muscles that help with support of the pelvic floor. Tose muscles are not in the abdomen so if your stomach area is hurting you probably are not doing them correctly.
Yes. Using any muscle you are unaccustomed to using can potentially make you sore.