No, there is no way for a VPN provider to prove they don't record any data. They have to process your original traffic at some point and can't prove afterwards that they deleted it without any copies.
Let's say you use the VPN to hide your communication with a plain HTTP website from your ISP. Here, the VPN doesn't magically create an encrypted tunnel between you and the website. It only adds encryption between you and the provider. Your request to the website will eventually leave the provider's infrastructure in its original form and could be recorded by anyone on the way. Since the provider has to handle your data, there is no technical way for them to not (temporarily) know it.
So, you can only either end-to-end encrypt or trust the brokers on the way. Other anonymity solutions such as the Tor network solve the trust problem by routing traffic over multiple nodes of which the majority is expected to be trustworthy and assumed to not collude with each other.
That said, just because they can't prove it doesn't mean there are no trustworthy VPN providers. When choosing a provider you should research their reputation, the jurisdiction they operate in and read about how they intend to keep your privacy and claim to handle your data.