You don't need a smart phone to utilize Strava in the National Bike Challenge. You can access a Strava account from a web based browser (I use Chrome) to log rides manually and/or upload rides from gps devices (I use a Garmin Edge 705 and Garmin Training Center). If you don't have a Strava account, the NBC registration process will take you there first. A free account is sufficient.
If you do have a smartphone, there is a Strava app that works with the free account on Android or IOS. From the app, you can record rides using the gps functions on the phone. When it works, it works great. But managing battery life and data coverage can be problematic and frustrating. From the app, you can also manually log rides. Support for all of these questions is available on Strava.