Anybody, even a manager, can REQUEST that time be called. However, it is entirely at the discretion of the umpire on whether play can, indeed, be suspended. For instance, if a batter steps out of the box and requests that "Time" be called, but the umpire thinks the batter is simply stalling, the umpire can refuse to call "Time" and instruct the pitcher to pitch. If the batter is out of the box, every pitch done by the pitcher is considered a strike.
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In Baseball, you can always ask for time. Unlike in other sports, teams do not "have" time-outs. Time is granted by the umpire at his discretion. As long as he feels that your request is reasonable and does not serve only to disrupt action, he will almost always grant time when requested. He may, if he feels a batter is slowing the game or trying to call time only to disrupt a pitcher, decline to grant time. The important thing to remember is that time is only granted as indicated by an umpire. Requesting time-out is nothing more than a request and unless granted by an umpire, base-runners are free to advance and pitchers are free to deliver pitches. The ball is live until the umpire indicates time.
Only the players on the field,the umpire,the coaches,and the batter or the runners. Nobody in the dug-out can call time out or the fans watching the game.
You may wonder why the players on the field or the runners/batter can call it is because if their shoe laces are untied and if they are hurt in anyway that the umpires and coaches need to know about if it's serious.