Is there a special time that the token shop restocks? Or is it entirely random?

No, there aren't any special restock times for the Token Shop, but it's usually announced in the news if the staff restocks it :)

Staff (mostly Keith or Amber) restocks it manually. If a new item is added, it will be announced in the news. Later on there can be 2-3 silent restocks of the new item, and sometimes silent or announced restocks of more (older) items.

Thank you! :)