No. You don’t need to upgrade to Pacer Premium membership to complete the challenge/race.
The registration code you receive upon checkout already grants you access to the challenge/race and the finisher medal, along with all the features you need to complete the challenge/race.
When opening the Pacer app for the first time, an "Upgrade to Premium" screen may pop up. You should be able to dismiss or skip that screen without a problem.
You are welcome to try the Premium membership. It unlocks many interesting features, such as training courses, fitness plans, advanced analytics, and more.