How can I buy from the iOS App Store and have it download to my iOS device?

The question sums it: How can I buy from the iOS App Store and have it download to my iOS device, perhaps telling it?

Anytime I attempt to go to the AppStore website, it keeps trying to push me towards iTunes, which I don't have installed on all my computers (e.g. my work computer). I'd like to purchase and push the app directly to one of my AppleID linked devices, it's gotta be possible in this day of cloud and mobile computing. (like Android or Windows Phone 8/10).

What am I missing here? Is there a different link? Different setting?