Could I take amino acids for recovery from training (wrestling, strength and conditioning) without gaining weight?

Yes. Amino acid pills or powders have a specific calorie content, but it is usually small. Amino acids are the building blocks of proteins such as those found in lean meat. Supplements are not likely to provide any benefit beyond a balanced menu including meat, dairy, eggs, beans and rice. I suggest you do a little research on real food sports nutrition. Gaining weight is NORMAL at 16, allow it!