I don't think you can sit at the actual bar. You can sit at tables with some moderate distancing.
You aren't going to find many "white sox" bars unless you go to the south side. I would advise just finding a bar/restaurant that isn't a big Cubs establishment. Most places will turn the white sox game on. Let us know what are you are staying in and we can give some recommendations.
Halligans will have the white sox on, but it can get pretty collegey. Not sure if it will be like that during covid, though. Divey Irish bar. I love the place. Actually saw Jon Lester there before. Kincade's is also a personal favorite of mine. Another divey sports bar.