Here are a few updates, and will add more as races make decisions about how to support safe events. In the meantime, you can find plenty of virtual races at RunSignUp, Running in the USA, etc.

June 2 The Big Run Chapel Hill
June 5 Running of the Bulls 8K Durham
June 20 Father's Day 5K Raleigh
June 26 Walk to Defeat ALS Triangle Area
June 26 Ultimate Runner Winston Salem
July 3 Cary Greenways Challenge Cary
July 17 Flip Flop 1K Cary
July 17 Beat the Heat 5K Winston-Salem
Sept 3 Rise 'n Shine Festival Pittsboro
Sept 6 Carying Place Labor Day Race Cary
Sept 18 Run the Quay Fuquay-Varina
Sept 18 St. Paddy's 4 Miler Raleigh
Sept 25 Pups and Pastries 5K/6.5M Cary
Oct 2 Cary Greenways Tour Cary
Oct 9 Oktoberfest 4 Miler Raleigh
Oct 30 Amberly Charity 5K Cary
Nov 13 Shakori 40 Pittsboro
Nov 20 Cozy Toes Raleigh
Nov 27 Turkey Chase Cary
Dec 5 Fortnight Before Christmas Cary
Dec 31 Last Mile Morrisville