Winner of my giveaway has private profile

The friend request button should still be there.

Sadly, this is the reason it is recommended to set a restriction level to your giveaways.

1. You can still add people with private profiles.
2. Private profiles are allowed, only, setting it on public is required to sync.
3. If all else fail, contact them by email.

Maybe try contacting him via email?
As for the private profile, it's not forbidden as long as you switch to public once a week to sync.
EDIT: Ninja'd :(

I completely forgot that i can login in steam on browser and add people. I couldn't add him on steam client because i couldn't find his profile. Thank you all for reminding me.

My bad. He activated game.