There are allergy tests for chicken; however, what you described was more like food poisoning. Most common form of food poisoning is from under cooked chicken. If you continue to feel ill, suggest you go to your doctor.
Although an Allergic Reaction may be the cause, a more common synopsis of the symptoms that are produced stem from food poisoning.
The symptoms can develop rapidly, within 30 minutes, or slowly, worsening over days to weeks. Most of the common contaminants cause nausea, vomiting, diarrhea, and abdominal cramping. Usually food poisoning is not serious, and the illness runs its course in 24-48 hours.
A recommendation would be that you wait the prescribed amount of time posted above, and afterwards purchase a pack of chicken (deli slices would suffice as well) and try a small portion.
Since it is precooked there is a very likely chance you may rule out food poisoning.
If the same symptoms persist, then yes although not entirely common, you may need to have an allergy test done to confirm whether or not you are allergic to chicken.
And as we are now on the subject of allergies, it is also advised that if you are experiencing adverse reactions to chicken, that you are tested for turkey as well, since they both have common properties.