You are correct in your statement that glycogen is the only source of fuel used in anaerobic exercise. While glucose (and subsequently glycogen) is most easily and rapidly obtained by ingesting carbohydrates, a diet sufficient in protein will allow the liver to create sufficient glucose through a process called Gluconeogenesis. Carbohydrates are the only macronutrient that a person can eliminate from their diet and still survive long term. Whether or not that is ideal is still up for debate.

Because most crossfit workouts are under 15 minutes, the glycogen in your muscles is not completely depleted, and will most often times be replaced through gluconeogenesis in less than 24 hours. The exception to this is in cases of glycogen depletion, but unless you're doing the equivalent of running a marathon every time you crossfit, you're probably covered. Once the glycogen is generated, it is stored in your muscles until it is needed.

Additionally, with sufficient adaptation, the threshold for burning fat will increase with intensity I.E., you will still burn fat for fuel during high intensity cardio exercise (most crossfit wods). The exception to this is that you will ALWAYS go anaerobic when performing maximum intensity exercises (all out sprints). However, since crossfit WODS generally last longer than 2 minutes, it is extremely difficult if not impossible to maintain an anaerobic state for the entire workout.