Keep in mind that no matter who authorizes, or how much of it you get from whom, nothing will protect you from litigation or negative reactions by people determined to go that direction. But, the fact that you made the attempt will stand in your defense if and when the poop happens.
That being said: I do a lot of youth sports video and this is my experience.
Schools: I get approval from the head coach. He or she gets what clearances they think necessary from the school administration, board, or parent/teachers group.
Youth leagues: I get hired on in a multitude of ways. I've been called in by a parent who wants video with particular focus on his/her son or daughter; I've been called in (or have approached) by the head coach, team mother or parent group to provide coverage, production and sales.
If there is a league board or local team governing organization I have advised my contact to obtain the necessary approval/clearances from them, and to determine if I am required to provide any kind of liability insurance.
I have also run into the obstinate bureaucracy of parent groups, officious league leaders and others who abuse their perceived powers, or something I call "parental politics".
My approach, legal or not so legal) is to request that the parent or individual get the necessary approval or clearance from whomever they perceive is the source for gaining permission. With that being accepted and established I have shot and produced for the coach, for the parent, for the athlete, for the team and/or for the controlling entity involved - board, etc.
Some have required I have a minimum liability insurance and proof of such; others have not. Some have required written agreements, others have not. I go with the flow, do not try to rock the boat and if the situation seems to be, or become, untenable I back off and seek video work elsewhere.
I also have done work guerilla-style, walking on field/sidelines and unobtrusively set up a shooting stand or ladder, whatever, and started shooting, handing out business cards (I even have a sign for the holder on my shooting ladder that says to inquire about copies of the game) or responding to inquiries. Sometimes I have been asked to leave, and I do so unless the venue and the situation is a public event where I have as much right as anyone to be present. Sometimes, if it appears the situation could turn nasty, rather than be right and aggressive about it, I go away - it simply isn't worth the hassle and bad press.
I have rarely been required to contact or go before the league (Little League, etc.) or to have special approval of showing league logos, etc. I have not yet incurred a legal situation in these events. It has been profitable and fulfilling for me as an event video producer, and I have the satisfaction of many groups and teams who request my return each season, as well as referrals for countless other video production work.
