I'm trying to add an alternate leader when purchasing some permits but I can only find a place for an email address. Why can't I add a name?

It's based on the email. Once you add the alt the system will send a email confirmation to the alt to confirm. The alt needs to have a Rec.gov account. Their name will appear once they confirm.

I received an email - Alternate Permit Holder Update - Your alternate permit holder request for Mt. Whitney has updated its status. Please check your reservation details for more information.So my reservation says Permit Holder Details: Alt1 Email: xxxxxxx@yahoo.com(pending). Does that mean my alternate leader needs to open a Rec.gov account?

Based on all the above, I would urge the alternate to register with that email address. They have probably received an email from recreation.gov, and it probably tells them what they need to do.