During the purchase, I don’t see why you couldn’t use your new email address.

Playing devils advocate here:
The only issue might be one of not being able to link it to your existing license, for any possible renewal discount and or the correct extension of your existing license expiry date.

But if the renewal is being done via the avast UI, I would like to think it wouldn’t be an issue. You could wait until closer to the expiry date or you can proceed. Should there be any issue with correct extension of the expiry date there are ways to contact sales to correct this.