Thanks for posting your question in the Apple Support Community. I understand that you're unable to use Touch ID to download apps on your iPhone.
First, I recommend to make sure that you have Touch ID set up to be used for purchasing/downloading apps by following these steps:
Use Touch ID to unlock your iPhone or make purchases
After you set up Touch ID, you can use Touch ID to unlock your iPhone. Just press the Home button using the finger you registered with Touch ID.
Make purchases with Touch ID
You can use Touch ID instead of your Apple ID password to make purchases in the iTunes Store, App Store, and iBooks Store. Just follow these steps:
- Make sure iTunes & App Store is turned on under Settings > Touch ID & Passcode. If you can't turn it on, you might need to sign in with your Apple ID in Settings > iTunes & App Store.
- Open the iTunes Store, App Store, or iBooks Store.
- Tap something to buy. You'll see a Touch ID prompt.
- To make a purchase, lightly touch the Home button.
Use Touch ID on iPhone and iPad
If you've set it up but are unable to use Touch ID to download apps, I recommend to follow the steps to Get help with Touch ID on your iPhone or iPad. The steps here aren't written specifically for this issue, but can help resolve this for you.
Let me know how those steps work for you.