In football, can the captain move players from one position into another, during the game, without the coach or manager telling them to?
Does the coach or manager need to give the captain permission for this to be done?
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Other than the goalkeeper, there are no positions defined in the Laws of the Game, so nobody needs "permission" from anybody play anywhere on the pitch. The captain, or anyone else on the team for that matter, can tell anyone to move into any position at all. Whether they'll be listened to our not is a different question of course, and that's down to the relationship between the manager, the captain and everybody else.
As stated by @PhilipKendall, there's no rule delimiting the functions of coaches and players. And to add to that: there's no rule that forces you to even have a coach present, nor for the coach to just coach.
In lower tiers, clubs with financial or management problems that can't find a coach, would play following the directions of the captain and/or the physical trainer. Take the example of Jumilla CF (Google-translated article), a modest Spanish "bronze division" team that's now in the top 3 of their group after surviving almost half the last season without a coach.
On the other side of the spectrum, a coach may choose to be a player himself too. When Zidane arrived to Real Madrid as a coach, there was some mild speculation about him actually playing some matches if he couldn't decide a lineup in time.
So much for rules about whether the captain can tell other players what to do :-)