Nope! Internet speed does not affect the performance of River Fox because the files are being stored and read locally. The only step that needs internet access is the licence check. As long as you’ve signed in and verified your account with the licence key, River Fox is ready to go.
If you are worried about your VR quality, make sure to check that you’re plugged into a power source and that you’ve set your computer's power setting to the highest performance setting.