It is possible yes, but it happens rarely. Generally a magical awakening is a latent birthright, something in your ancestral bloodline, but there have been anomalous awakenings, usually either from surge metavariation, time in a magically active area or around magical individuals, or some kind of pollution(radiation, magically active dust particles,etc.) In GM's terms it is handled as the player paying 2x the qualities cost as karma(if the character doesnt have karma, just give them a negative pool). The only downside is that until trained either through personal study or the intervention of a pc or npc, it is difficult for this talent to become anything close to what awakened are accredited with. they dont have a bp system for 5th edition rules yet, but im sure that the doubled cost of any magic quality of 4e would translate nicely, might even be a bit more of a penalty than it was in fourth to their karma.
The only other way It could occur, is through hmvv, or any other method of becoming a dual natured being.
would the player enjoy being awakened is the bigger question.
