How do I purchase an app for someone else to download to their phone?

You don't.

You need to use their iPhone and their Apple ID to download/install any app that can be used on their iPhone.

or

you can use this procedure:--> Send a gift from the App Store, iTunes Store, Apple Books, and more - Apple Support

Regards

Giulio

Send a gift from the App Store, iTunes Store, Apple Books, and more - Apple Support provides the requirements.

Give them the money it costs.

Oops sorry Ferd :) Didn't see your reply in time.

Giulio

😎