Can I still have esophageal surgery even though I have a mild case of bronchitis?

Maybe... It depends on your symptoms and oxygen level on the day of the elective procedure. The surgeon and anesthesiologist will make the decision on the day of your surgery but if it is delayed, they have your best interests in mind. You don't want to take any unnecessary risks!
Best to wait. Untill after you have recovered from the acute episode of bronchitis.