Is it possible to have a yeast infection with only burning and itching?

Yes: Yes it could be yeast infection.

Yes: In women, yeast infections are the second most common reason for vaginal burning, itching, and discharge. Yeasts are found in the vagina of 20% to 50% of healthy women and can overgrow if the environment in the vagina changes. Antibiotic and steroid use is the most common reason for yeast overgrowth. But pregnancy, menstruation, sperm, diabetes, and birth control pills also can contribute to it.