Can my child get strep throat without having his tonsils (removed 2 years ago)?

Yes !! Strep pyogenes (group a) does not care whether its victim has tonsils or not. Not only can your child get strep throat, but this should be tested for and treated, because without eradication there are post-streptococcal complications like extension of the infection to other areas, or acute rheumatic fever. Get child seen by family doctor.