It will work fine.
EF lenses will work on EF-S cameras, but not the other way around.
As for will the 70-200 work well for YOUR sports event, that depends on the event, and where you are vs the players.
I know a yearbook advisor at a high school with T7.
To me that camera and use is a mistake, because the T7 has a max ISO of 6400.
This is a problem in exposure in LOW light.
If I am shooting at ISO 12800, f/2.8, 1/1000 sec.
With the T7, the max ISO is limited to 6400, so my only option is to shoot at a slower shutter speed, and risk motion blur. ISO 6400, f/2.8, 1/500 sec.
A volleyball spike cannot be frozen at 1/1000 sec, I tried. The blur would be worse at 1/500 sec.
But if you are shooting diving, 1/500 sec will probably work fine.
This is what I meant by "work well for YOUR sports event."