Is it possible to have an asthma attack/along the lines of, from getting your ears pierced?

Not directly. The piercing itself won't cause an asthma attack, but anxiety is a very common cause. If you're very nervous and apprehensive about it, try to calm yourself down and be sure you have your albuterol with you. If you can't beat the anxiety, maybe piercing isn't for you.
Asthma. While many things can precipitate and asthma attack, getting your ears pierced is an unlikely cause.