You can eat deer at any time of the season. However some deer populations have "wasting disease." So, you can have all of my share. The musk gland is not affected by the climate, or season.
As soon as humans figured out that you shouldn't eat diseased animals, they also concluded that after winter arrived it would kill off all of the sick and weak animals, so people felt more comfortable with winter kills than with summer kills. Also, deer breed in the autumn and give birth in the spring. By not killing deer in the summer you are not depriving a fawn of its mother, it will be on its own by winter.
All of these factors have entered into our belief systems and consequently, some people believe that killing deer in the spring or summer is some kind of food taboo. No, it is just common sense. (codified into law in most places)
In every walk with nature one receives far more than he seeks.
--- John Muir