"Hey mom, there's a summer camp I'd really like to go to this summer, can I please?"
show her their website, explain that you'll have a great time and it'll be a great experience, say that it's only 2 (or 3, or 4) weeks. tell her how it's a reliable experienced camp and the instructors know what they're doing and show her testimonials on the website if there are any.
if it's too expensive, look for a cheaper camp or apply for financial aid.
be polite and good-natured when asking her. if she says no, don't get whiny or upset, calmly ask her why.