No, you do not need an ID to buy groceries – for the moment anyway. But some states require you to produce a specific ID card for you to exercise your constitutional right to vote.
One of my former hobbies was registering people to vote. I’d do it at street festivals or events and I’d even bring my stuff to parties. It was cool to register first time voters.
So, I have an opinion on voter ID laws, which are just an attempt to suppress minority votes and votes of other economically vulnerable people, who may move often and can’t afford $20 for a government ID each time they relocate. The discriminatory intent of these ID laws is plain to see when you notice they usually allow NRA membership cards but not college IDs to vote.
A person establishes their eligibility to vote when they register to vote, not when they attempt to participate. To register to vote, you have to establish your identity and address. Just like a state ID card, you do this by providing a postmarked utility bill or some other official correspondence received at your current address along with a birth certificate or some type of a photo ID.
You attest to your eligibility to vote when you register and swear under penalty of perjury that the info you provided is correct. When you go to vote, a copy of your registration is there that contains your signature. If a person’s address and signature match, that person should be allowed to vote. Voter ID laws are a thinly disguised attempt to suppress minority voting.
Here are Illinois requirements to register to vote.