Can you have an allergy to insect bites?

See below. All insect bites can cause local reactions which include redness, swelling, warmth and itch and this is normal bit sometimes, a person can have what is called a "large local reaction" which is a very vigorous version of the local reaction but is not necessarily an "allergic reaction". Some people can be very allergic to bee stings which can cause anaphylaxis. As for other bites, it is possible.
Yes. There are numerous insects to which people have been shown to be allergic. Bees, wasps, hornets, ants, mosquitoes and other insects can trigger your body to produce ige, and once it does, the next time you encounter the insect, you may have a reaction.
Yes. Biting insects like mosquitoes and gnats usually cause swelling, redness and itching at the site of bite. However, stinging insects like honey bees, yellow jackets/hornets and wasps can cause life threatening generalized allergic emergencies.