How would you handle it? Some questions to consider:
Would you use games or innings?
How many would be required for a player to qualify?
Would he have to play the qualifying number of games or innings at the position he is being judged at, or just in general, and then he would be evaluated at his primary position?
Regardless of how you answer the third question, I think outfielders would have to be treated the second way. Mike Trout played only 885 2/3 innings at center field, but 1,225 2/3 overall. He clearly should qualify.
Once you've determined your qualifications, I'd like to see your choices for the awards at each position, AL and NL. Pitchers optional.