In the poem "The Blue Bowl" by Jane Kenyon, what would the rhyme scheme be? Is there a rhyme scheme?

It is true that there are no end rhymes in this poem, nor is there any discernible metrical pattern.  The poem therefore is indeed free verse, although you are correct in noting that there are a couple rhyming pairs.  There are the words “bowl” and “hole” in the first four lines, and “toes” and “nose” a few lines later.  It would be safe to say that these words constitute two instances of internal rhyme, words found within the body of a line that rhyme with each other, rather than a traditional end-word rhyming pattern.  This gives a poem an extra layer of verbal harmony, and displaces the traditional end-heavy focus found in traditional poetry.  This poem has a further disruption, as well,  made possible by the interesting free verse structure.  By ending phrases in the middle of lines we do not read the poem as we typically would, from  line to line, each one encompassing a single thought.  Rather, we read it more as a continuous story.  By breaking up the structure of the poem in this way, Kenyon is playing with the definition of the poem itself. 

In addition, something worth noting is that, had each line of the poem been dedicated to a full phrase, these rhyming pairs would be caught at the ends of their lines.  However, as it has been arranged, the poem has them placed internally.  By doing this Kenyon has avoided all end rhyme; there is therefore no true scheme for the poem, but two isolated examples of an internal device.