What is the relationship between radiation and evolution?
Is background radiation a critical component of evolution? No, it most certainly is not.

The DNA replication and DNA repair mechanisms aren't perfect and errors happen without any external cause or catalyst. You could say mutations happen on their own. There are mutagens that also cause DNA damage or mutations, but they're merely affecting the DNA replication in a negative way, causing it to err more often.

So, the effect of background radiation isn't the key here, the key is that the biological processes of DNA replication and repair are not infallible, although radiation does increase the error rate (the rate of mutation) just like many other mutagens do.