Hi 15 year old male here. I'm 5 8 but my wingspan is 5 7. Does this mean I'm done growing. I started puberty at 13?

I think its unlikely. Almost every guy in my year at school had a random growth spurt last year (thats year 11 - i think its the same as 10th grade?) so in a year or two you probably will grow a bit more. 15 is quite young to stop growing so I wouldn't worry about it if i was you; theres nothing you can do to change it anyway.