The answer to this question is very subjective, and highly dependent on your definition of "Kingdom of God," especially as it pertains to Earth. Some might cite the verses in Matthew or Luke, known as "The Beatitudes," as a point of reference:
Blessed are the poor in spirit, for theirs is the kingdom of Heaven (Matthew 5:3).
This verse is mixed with verse 4:
Blessed are those who mourn, for they shall be comforted (Matthew 5:4).
Based on these two verses, however, some look at the moments of mourning which take place in the film (Joe's desire to know his father, Rebecca's death, the bus accident, Simon's last goodbye) as his experiencing the kingdom of Heaven on Earth.
However, I think if you look at the verse as it is found in scripture, you can make arguments that Simon is also poor in spirit, which could mean lowly, humble, and/or submissive. In many ways, Simon cannot control his humble position in society. His physical size is the first thing that makes him beneath others. Also, his parents are not only poor, but they do not develop relationships with their son. He seeks love, attention, and maternal affection from Rebecca.
Throughout the movie, Simon talks about his "fate" or his purpose in life, and that he knows he is supposed to be a "hero." However, he only talks about this with Joe. He never proclaims it to anyone, and he never flaunts this knowledge over anyone. He expresses it as though it is simply his humble duty.
Therefore, when he walks with his best friend through the death of his mother, leads his best friend to know the identity of his father, and saves the children on the bus, he believes he is only doing what he was put on Earth to do. According to the definition above, these would be times of Simon Birch, through humility, experiencing the kingdom of God on Earth.