I'm from FL, want to buy groceries for my cousin that lives in IL & have her pick it up, how do I do that?

I'd recommend a Walmart gift card. You can buy the card and give her the details. Walmart gift card options are available at: https://www.walmart.com/cp/gift-cards/96894

Thank you Ken Bass and Wilbert Jenkins for your advice

You can go to your local Walmart and send money to her Walmart. She can get it at her local Walmart.

You may be able to do that now. This store just started online Grocery pick up. If it's anything like site to store, you might be able to buy online and have an alternate pick up person. But I haven't tried.