I predict that the needed self-confidence will come with time and your patience. Don’t scold her if she suddenly clams up on you when out in public. Unless there is something really critical in the situation, it is probably best to pay no attention to it. If someone speaks to her and she doesn’t respond to a compliment or question, just casually remark that she is shy around strangers. There is nothing wrong with having her hear you say that. The soccer camp may help, but it may also overwhelm her.
About the music: we know that Mozart was composing at age 3, but that degree of precocity occurs only about once every 300 years or so. Most piano teachers prefer to wait until children are 6 or 7 before beginning formal lessons. The hands of very young children are so small that it is difficult for them to do much at a standard keyboard. Also, in order to read music they have to realize that the little characters appearing on or in the different lines and spaces stand for specific notes, just as in learning to read words they have to grasp the idea that the different squiggles of print stand for specific sounds and words. It is easier for children to do that after the age of 6.
If you already have a piano, it is a good idea to let her sit on the bench, press the keys and produce whatever sounds she can, provided you can stand it. There is very little likelihood that she will damage your instrument. And you can play very simple tunes for her, like “Mary Had A Little Lamb,” and note whether she tries to copy you. If so, find a good teacher and forget what I said about waiting until age 6 or 7!
Finally, I have to mention the Suzuki violin method as a way to begin teaching music to your daughter, even though you indicate that you are primarily interested in the piano. The Suzuki method teaches children as young as 3 by using small-scale violins. The instruments have to be exchanged for larger ones rather quickly, but it is usually easy to work out a trade or sale to other interested parents. Except for toys, scaled-down pianos that fit young children’s hands are not generally available. If you feel that your daughter is ready to learn a musical instrument now, I would look into whether there is a Suzuki school in your community. And don’t neglect singing. We carry our “vocal instrument” with us through our lives. It comes at no extra cost and won’t ever have to be traded in!
Our parenting advice is given as suggestions only. We recommend you also consult your healthcare provider, and urge you to contact them immediately if your question is urgent or about a medical condition.