If I'm 19 and got a consumption by minor ticket last week, will that affect my chances of receiving my work permit?

I suggest that you consult with a criminal defense attorney with respect minor consumption of alcohol ticket you received to see if possibly the court system where the action is venued has a "diversion" program that you can be referred to. If such happens, then if you complete the program successfully with paying a fine, doing community service work, and staying out of trouble then the charge could be dismissed by the court without a conviction.

If such happens you would not have a criminal conviction for a background check.

Since you are an adult (age 19) you do not need a work permit.