Chicago and most of the suburbs are very dog friendly. We have tons of events here in the warmer months, such as festivals, outdoor concerts, fairs and parades and many of them allow dogs although there are a few that don't... Chicago has tons of neighborhood and city festivals so there are a lot of options, I usually have certain events I try to go to with my dogs every year such as Custer's Last Stand festival, Long Grove Strawberry Fest (and/or their Apple or Chocolate Festivals), Midsommarfest, Northalsted Market Days, Chicago Korean Festival, , St Charles Scarecrow Festival, Chinese New Year Parade, etc... I've also taken my dogs to the Movies in the Parks which are different movies that are shown all summer in different city parks.
I've never been asked for proof of vaccinations when going to a public type of event, only when going to a dog show or pet show. I don't generally carry that around with me although I do keep a copy in the car and of course my dogs wear their rabies tags which are required by law to be worn.
There are also dog-specific events and festivals such as the various rescue and shelter walks/runs which usually also have other events that go on before/after the walk itself, like costume contests and for-fun agility courses and vendors and things like that. There are also several dog-specific festivals like the Dog Day on the Green or Woofstock, and then there are the pet shows and dog shows of course, and a number of fundraisers for different rescues and shelters that you can bring your dog along to. Many restaurants also allow dogs on the patios although in the city proper they actually need a special license for this so it's important to check first if they're allowed (in the suburbs they don't need a license).
You could probably take your dog to a different event here pretty much every weekend day in the summer, and still not get to everything that is dog friendly.
Here's a photo of Bianca from the Long Grove Apple Fest last year: