So the giveaway winner hasn't claimed the code yet and it's been a week.

Check his library, check if he redeemed it, or contact support :o

Try resending the mail in case it was marked as spam or try to add him on Steam.

Send them a friend request and see if they have seen it yet.

In future, best to give them the key over chat since you can confirm they have it, since rerolls can't be done with a key if you have emailed it (unless you want to risk them having used the key elsewhere, leaving you with a new winner that won't receive their game unless you purchase a new copy)

This is why emails are no good. You can't tell if they would get it just fine or not receive it.
I would actually try contacting the person and see if he/she has received it yet.

Can you re-roll at this point?

just provide proof to staff that you sent the email and have them mark it as sent for you idk if this would work, but it seems reasonable to me