If you do not, in a few years you'll have mint growing absolutely everywhere.
Mostly kidding, I grew up living at a rural property that had mint all over the place. I loved it, but I do understand how much trouble invasive plants can cause.
Here's some links I found for ground cover plants good for clay: https://gardening.stackexchange.com/a/7665 https://gardentabs.com/ground-cover-plants-for-clay-soil/
As for where to get stuff, search for plant nurseries near you. They may not be tech-savvy so you might have to use the old-fashioned technique of wandering around and asking people (anyone who works in dirt: farmers, gardeners/landscapers, contractors). They will know your local environment and what works best for it and can order plants and seeds for you.
AeroGarden is a pricy brand, but there are probably cheaper ones that do just as well.
On top of that, variants of thyme work as a hearty no-hassle replacement for grass, and it even flowers.