After what time would it be appropriate to request a new winner?

just do it now ;)

Yes, you can ask support for a reroll if a week has already passed.

Okay then, thanks for the lightning fast replies :D

1 week

7 days - always.

After 604800 seconds

Make sure you've tried to contact them in several ways (e.g. email, Steam) and have evidence of that.

if you tried to contact the winner (friend on steam or email), just send a reroll ticket.

Legally, one year + 24 hours

or after one week + proof you tried to contact him/her

After 7 days.

like mully said, support will want proof of a contact attempt outside of steamgifts. doesn't really matter which you choose just as long as there is/was an attempt to notify outside of steamgifts.com also.

so 7days, +contact outside of steamgifts.
in my case i tend to wait the 7days, email/notify the user and give them an extra 2days, then fill out the ticket. but you could do it however you like as long as it's after 7days and you have some proof of a contact attempt.

Actually they didn't ask for it. I told them that the winner hasn't been online in a week and they just rerolled it.