MyChart is a method many doctors use for patient communication. It makes it easy for patients to send questions, get lab results, the doctor to order labs, etc. If you aren't familiar with it, your doc probably isn't on the system. An alternative is the older method of calling the doctor's office, telling them you have colored mucus and ask if they want cultures. It's really up to you if you think you need to call. Doing it before appt would just speed things along. Good luck to you.
Edit to add: One additional suggestion is to ask your doctor at your appt for a standing order for a sputum culture. My pulmonologist keeps an order on file in lab and sends me home with a specimen cup kit so that if I start getting up sputum that is suspicious, I can send a specimen in ASAP to avoid delays in treatment.