Is it possible to develop food allergies that you have never been allergic to before? I'm 24 and have a reoccurring rash due to something I eat.

Certainly. Allergy reactions require that you have had a prior exposure; you can't have an allergy to something you have never been exposed to. Allergies, including food allergies, can appear later in life. You should find a good allergy doctor and get tested.
Most common foods. Which cause issues in adults are peanuts, tree-nuts, shellfish, and fish, but certainly any food allergy can occur at any age.
Also. In addition to dr. Rezvani's answer, do consider celiac disease. In some people the sole symptom - ie the one that they notice - is a rash. The rash has the very unglamorous name of 'dermatitis herpetiformis'.