I am writing a children's story in first person, present tense. Aside from the quotes from characters, all of the text is meant to represent the narrator's own thoughts, and either describes what they see, do, or think, or their observations on other characters. I found that some areas seem more natural when I use contractions. Would it be appropriate for me to use contractions throughout the narration? Are there any authors who have done this successfully?
Contractions are fine in narration. As with other aspects of the narrator's diction and voice, the use or avoidance of contractions helps characterize the narrator, and indicates something about the formality/informality of the story.
If you are writing in first person, the language used needs to be roughly mainstream consistent with the age, location, etc. of the narrative character. If the two are hugely out-of-sync, it can cause a lot of discord while being read because the "person" that is speaking is saying things and using words that are out of character, thus unexpected and often jarring.
That said, when writing books for young children (pre-teen or early-teen), you have the additional challenge of the fact that the people buying the books (parents or schools most often) want the kids to pick up good language and grammar skills while reading, instead of reinforcing bad ones.
Contractions are not typical for a very young child but you could get away with them if your narrative character is a teen. A six year old using a contraction "isn't" wouldn't ring true to me and I wouldn't expect my six year old to know how to properly use it. It's not as bad as a ten year old using the word "transcendental" but it would still jar.
Here are two potentially useful links:
Note that the contraction worksheet is a 5th grade worksheet. That translates to roughly 10-11 years of age in the US. (My references are US references, I don't happen to have links handy for UK or other countries).
This page seems to suggest that books that teach contractions target children between the ages of 4-8, so it would depend on what age children you're targeting, it could be that the average 4 year old might not understand contractions however I would very surprised if an average 8 year old didn't understand them. Otherwise, they're proper grammar so if you expect your target audience to understand them, by all means use them.
I would add that, as someone has already mentioned, a first person narration should be written in a similar way to the narrator's natural way of speaking.
An additional consideration may be whether you want your children's story to be stocked in school libraries.
A colleague wrote a piece of Young Adult Fiction intended to encourage reluctant readers but found it was rejected by schools because of "the awful grammar"! It was not considered to be a redeeming feature that the main character's expression changed (improved?) as he incorporated what he had read into his own thoughts and speech as the story progressed.