Hi Savannah,
 
Typically, study abroad programs are geared toward high school and college students. While there are some travel programs for younger grades (through schools, churches, community groups and travel organizations), those typically involve parent or adult chaperones, and are generally short-term trips (like field trips) rather than year-long, live-in placements.  
 
While I can understand your fear of your mom rejecting your goal to study abroad, I still encourage you to discuss this with her. She might even enjoy doing the research with you!
 
Here are some resources the two of you could review together in planning a study abroad trip once you're old enough:
 
Education.com Study Abroad info center:
http://www.education.com/topic/study-abroad/
 
StudyAbroad.com
http://www.studyabroad.com/
 
Here's an article you may find helpful for discussing this study abroad goal with your mom...
 
Persuading Your Parents: You Want to Study Where?!
http://www.education.com/reference/article/Ref_Persuading_Your/
 
In the meantime, you can still learn about Japanese culture without even leaving your home. For instance, you could check out this website, Kids Web Japan (from the Ministry of Foreign Affairs of Japan): http://web-japan.org/kidsweb/  
 
You could ask your mom to rent Japanese films, such as anime. Here's a collection that Netflix offers (they deliver the movies to your home via the mail, and your family returns them the same way): http://www.netflix.com/Search?v1=japanese+anime
 
Or you could try cooking Japanese food (http://www.japan-guide.com/r/e1.html ), or practice the Japanese language (http://www.japanese-online.com/ ).
 
These are just some of the ways you could enjoy Japanese culture. Finding a Japanese cultural center or events in your area, or visiting an Asian art museum, are some other ways you could explore Japan locally until the time is right to take the interest abroad.