Yes you can be allergic to processed meat.
Processed meat can contain preservatives as well as nitrates which many people can be allergic too.
The most common reason people are allergic to processed meat is because of the nitrate and the person having a nitrate allergy.
Nitrates can cause itching and rashes in people allergic to nitrates and when they eat that meat they can get that itching and skin rashes.
The meat allergy or nitrate allergy may also cause stomach pain, diarrhea, bloating etc.
Processed meats are high in nitrates and nitrites.
Some people also just have a harder time digesting meats than other people and eating meat when you have digestive issues with the meat can cause bloating, gas, stomach pain, stomach cramps and gas.
If you're allergic to meat then you can eat vegetarian meat that is made from Soy or other non meat foods.