My Child is going into 8th grade and she feels like she is doing the same work every year. She wants to go to Fletcher Academy, but can't get accepted unless she has proof she is able to comprehend what they are teaching. I have no clue how to get her tested or where, but I really need to get her tested soon.

Ask Fletcher Academy who does their testing.

Seems like the best way to do this is to call Fletcher Academy and explain your situation:

P.O. Box 5440 | 185 Fletcher Academy Drive | Fletcher, NC 28732
Phone: (828) 209-6800 | Fax: (828)209-6809

You are asking two separate questions, one was how to get her tested to allow her to skip a grade,

The other question was how to get into this academy ( I shall assume a private school of some type)

For the academy, most have their own entrance exams and/or will look at current grades in the current classes they are taking.

For skipping a grade, again each school will have their procedure ( or no procedure and will not do it) Normally they use grades from current year and test using the End of the Year Test, ( which ever is used there, like the CRCT or the CAT testing