My son has allergies that effect him in many different ways (nasal allergies, allergic croup, asthma). He has seen an allergist, a pulmonologist, and an ENT for different allergy related issues and all have been helpful. In his case it makes sense to use the allergist as his primary person since she is the only one of the three who can treat all of his allergies (ENT doesn't really do the lung issues, pulmo doesn't really address the throat and nose issues). However, we love his ENT and would be really happy with his treatment if his issues were limited to Ear Nose and Throat.
If your child's allergy issues are primary upper respiratory then I'd probably call the ENT's office, and talk to them about whether they feel comfortable treating them. If the allergies are different (e.g. asthma, eczema, anaphylaxis) then I'd think an ENT might not be the best choice.
By the way, my son's issues have never involved his tonsils or his ears -- ENT's do far more than those two things.