I think some things are just common courtesy. It has nothing to do with bowing down to a coach or asking them to approve of your decision. Likewise owners and coaches must understand that they have to work every day to keep people coming to their gym and should not keep people that do not want to belong to their gym - even if they don't want them to leave.
Even if you leave at the end of the year, many coaches are ALREADY in planning mode for next year. They are already thinking about whom they may like to try in certain spots and put in a different position. They are already in their minds if not on paper thinking about next year teams. Letting them know as soon as you know lets them re-adjust their thinking so they are not counting on your son or daughter to return when you know for sure they are not. They may not plan to do that Jr 4 or Senior Open 5 team if a certain amount of kids inform them in advance they are not coming back. Of course you don't know who may come in the door but you can never count on that.
If you leave do it gracefully and in person. As has been stated gyms close, merge, and partner with other gyms. Coaches move from one gym to another. You may have to deal with the same owners and coaches again. It is always best to leave at the end of the season. But if this is not possible, it must be with a personal conversation with the coaches/owners. Not a telephone call, e-mail, or letting it be found out through social media that Suzy is taking her tic toc to another gym. Be upfront and communicate. Leave any and all concerns all on the table and if there is a balance due pay it before you leave or make suitable arrangements and get it in writing.
When you leave, go. Seriously. If you do not want to be at Gym A anymore fine. But don't act like you still want to be there, want to know everything that is going on, how the new teams are looking, where they are competing, etc. That bugs coaches more than they will ever tell you. You can be cordial and respectful to them without being nosy.
Say no to the temptation to try to pull other members with you because your new gym is offering a special discount to you and them if you do. This is a very old recruiting trick and most gyms are aware of what their competitors do to try to undermine their business. Don't become a pawn in that game. It will get real ugly, real fast and you will get caught up in it.
Remember that every gym has a tattletale, so be upfront and honest about why you are leaving. When you go to your new gym and tell them why you really left your old gym, it will get back to your old gym. It happens every time. This will always generate bad feelings. especially if it is a different reason from what you told the owners and coaches when you left.
Coaches and owners - let them go. It is hard when you have invested so much of yourself into an athlete and they or their parents treat it as a mere business transaction. It hurts even worse when they hardly acknowledge they learned anything at your gym when you know that if it wasn't for the training they received at your gym, they may not have made their dream team at the new gym. But clinging onto the rejection will only make you bitter in the long run. No matter why they left - whether true or false - Let it go. Take a deep breath, scream in the woods if you must, then double down on your efforts to continue to grow your program with those who believe in you and want to be a part of what you have to offer. If anything make your program such that they wish they never left.:)
If they come to you and shoot straight, offer them your support and even contact the new gym to let them know that they are on the way. If coaches take that step to at least communicate that type of movement it will do a lot to diffuse drama. Again you do not have to be the best of friends to call and say Suzy is leaving our gym and says she is coming to you. We wish her and you the best. Do not give the new gym any bad reviews even if asked. Let the new gym find anything negative about them on their own.
Communicate your vision! This is the biggest area where many small gym owners and coaches "lose it." The Nehemiah principle states that vision must be recast every 52 days. While you may not want to go 52 days at least once a quarter they should hear directly from the owners as to the vision of the gym by some format - letter, meeting, focus groups, website, e-mail, etc. Too many owners just assume because people walk through the door and smile when they are there that they will be there forever. Or think that because they were a good cheerleader, they will be a good owner. Don't ever get so busy that you can tell those who pay you $$$ and those you pay $$$ what the plan is. Let your parents, athletes and staff know the plan, signposts they can look for on the way, and by all means allow them to get off the train if they are not comfortable with your vision. Make sure your coaches and teachers are effective and on your page. Get rid of dead weight - staff, classes. programs and sometimes students. BTW "to win" is not a plan - it should only be a signpost on the journey to a greater destination.