E.g. the possibly most prevalent allergies, against pollen, manifest well after kindergarten, or even in teenagers. Until then the system did not care at all about pollen. The reason must be something entirely unrelated to pollen. And there are more and more cases of people developing pollen allergies well after age 50! (But: often you will find other allergies or "pre-symptoms" in kids, e.g. neurodermatitis or milk-protein allergies).
In fact you always develop the allergy on exposure AFAIK - it just might be the first exposure. But the reason is most often completely unrelated to this specific substance/protein.
There are even several known substances that are known to cause allergies to develop on exposure (with high probability). But what you become allergic to is pure chance, e.g. could be the nickel-plated jewelry on your skin which you never had a problem with until then.
I have allergies to shellfish and tree nuts (plus celiac disease - gluten) diagnosed/developed later in life, along with oral allergy syndrome which sometimes causes a painful reaction to a wide range of foods, but only sometimes (depends on immune state). While I’ve never had an anaphylactic reaction, my allergist prescribed an EpiPen, saying the reactions often increase in severity suddenly with no warning.