Ren-chan's "Frog's Song" is either a parody or otherwise an original song.
As nhahtdh commented, the song lyric couldn't be found outside of Non Non Biyori-related results, making it an original song. The song is also known to the fans as れんちょんのかえるのうた (Ren-chon no kaeru no uta, Ren-chon's "Frog's Song"). It can be listened on Nico Nico Douga, starting at 0:45.
Frogs are grown-ups
Tadpoles are children
When they grow hind legs they are moochers
When they grow front legs they are rebellious
On the other hand, there is also a Japanese nursery rhyme called かえるのうた (kaeru no uta, Frog's Song), also known as かえるのがっしょう (kaeru no gasshou, Frog's Choir). The song can be listened on YouTube. However, the melody and the lyric are different.
クヮ クヮ クヮ クヮ
The frog's song
We can hear it
Ribbit ribbit ribbit ribbit
Ribbit ribbit ribbit
Lastly, there are other songs (collected on Nico Nico Douga) Ren-chan has sung on the anime regarding rabbits, little birds, and dynamites, nekojyarashi (cat toys), paint it black, a fake doctor and a ripped love letter, and a diary. Considering the nonsensical lyric and sometimes heavy topics, all of these don't seem to be taken from existing Japanese nursery rhymes; parodies at best, or else original songs.
Considering all of these, Ren-chan's "Frog's Song" is either a parody of a famous Japanese nursery rhyme with the same name or otherwise just an original song.