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Charlie,

They no longer have access to their old email account, and they have
never signed in to the GIO website (and hence no saved authentication
cookie).

Nor a password to log in with, if they never set one up.

... is there any way they can change the email address associated
with their GIO account?

Not if they can't log in to Groups.io, nor receive any messages sent to the old address. To log in you need one or the other: a password in your Groups.io account or access to your email inbox.

Assume that at least one of the groups to which they are subscribed is
a free (non-premium) group.

As Bruce mentioned, the more useful question is whether he/she is subscribed to one or more Premium groups. If so a premium group owner/mod can change the email address for him/her. That will switch all of his/her subscriptions to the new address.

Shal


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