The nearest charter school is Weilenmann School of Discovery (their website is: http://wsdpc.org/). It's about 15 miles from Deer Mountain Blvd going towards Salt Lake City (near the Parleys Summit). The school's address is: 4199 Kilby Rd Park City, UT 84098 and phone: 435-575-5411
I live about a half mile from it and it's a good school. Acceptance into the program is based on a lottery system -- the website has more info on the lottery. But, enrollment for the 2013-14 school year began in January with lottery drawing once a month February through May. Classes for a given year group are limited to 60 to 75 children. They also accept children from outside Summit County/Park City.
Hope that helps answer your question ... let me know if you need any more info,
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