As a piano teacher, I have had people ask if it is possible for them to learn piano on their own. I've also had parents (especially homeschoolers!) ask if they can teach their children how to play if they don't play, themselves. You may have to be a homeschooler yourself to understand a question like this, but we are a strange breed, and always think of doing it ourselves, first! This question doesn't have a simple yes or no answer, so I wanted to draw your attention to an article I wrote on my website, "Music at Home". It is called Can I Teach Myself To Play An Instrument? I hope it clears up some of the questions you may have!
For the record, I do encourage you to give your children the opportunity to learn an instrument. If you don't know which instrument to try, read this article. It gives ideas for choosing the right instrument for your child.
Summer vacation is a great time to try something new, and some parents like to start music lessons while their children don't have as many other things to worry about. However, there is no best time to start. It really depends on the kid. I actually made the mistake of starting my son's piano lessons too early. He seemed interested, and he seemed to be able to understand it all, but was tortured by having to practice every day! We ended up quitting for a while, and he is doing much better now. So, don't be worried if you try an instrument and it doesn't work out. There's plenty of time. Waiting a while or trying a different instrument might make all the difference.