Sure! People do it all the time!
We recommend this for folks who have a few wigs in their hair wardrobe, though. If you only have one, you may just want to throw a ball cap on instead and spare your hair from wear and tear* until you can afford to rotate them out.
Most women who do this have a dedicated “workout wig” that they only wear for workouts. We recommend something lightweight and open-wefted so it breathes well while you sweat!
Another great alternative would be something like a ball cap with hair attached. These are very easy to work out in, and often much more comfortable during outdoor workouts than full-cap wigs.
Toppers can also be of utility if you have enough hair to support one. Put a topper on, pile your hair up in a ponytail, and away you go! ;)
* Poetry! :D