Yes, even if the app is free to download, you may still need to have a valid payment method as Apple says in step 2 of this Support Article:
If you can’t download or update apps on your iPhone or iPad - Apple Support
I do know that states require Apple to collect taxes on their purchases based on the region that they live in. Some users have attempted to change the location of where they live to avoid paying taxes on purchases in states that have a Sales Tax. The payment method must match the region that is set on your device in order for it to be valid. With that said some users have been able to download without a payment method, but if you are met with the message that you need one, then there is no way around it.