I have done grocery shopping and even Costco runs on my scooter - no joke. My wife and I bought "clearance" aluminum racks from Target. The one photoed here is a small one and cost me $7. My wife bought a huge one, that cost her $10. Many people just use plastic crates. I wouldn't count on bungee-cords . . . rather I would hit your hardware store up and buy 4 strips of "framing" metal and some bolts, nuts, split washers, and flat washers to sandwich and lock the basket in place.
I chose use hardware that allowed to remove my basket without tools any where I went. It cost me about $10 total, but you can do it for a lot less if you wish. I tend to over do things. My basket was extremely sturdy and mounted to my rear rack on the BBQ grill.
I held everything in the basket with a bungee-cord net.
Amazon sells similar baskets for $12.