"The Little Vagabond" uses repetition to capture the little speaker's chirpy voice (and his comical conviction).
Take, for example, the epizeuxis in line 1:
Dear Mother, dear Mother, the Church is cold,
These two "Dear Mother"s come in quick succession, like tugs on a sleeve, and establish the child's urgent, pleading tone. He is supremely uncomfortable in the church, and really needs his mother to hear his bright idea for solving this problem: make the church more like a pub!
He gives some of his reasons for this plan in lines 3-4, using some emphatic polyptoton and epistrophe:
Besides I can tell where I am use'd well,
Such usage in heaven will never do well.
The polyptoton on "use'd" and "usage" sets up a clear contrast between the way the pub and the church "use" (or treat) the boy. Similarly, his epistrophe on "well" (which also creates a bold identical rhyme) stresses that the church isn't treating anyone "well" at all. Rather, it's cruel and repressive in a way that the speaker can't imagine would go over "well" in heaven.
In line 7, he pictures the much more cheerful scene of a church that had learned some lessons from the pub. Here, the parallelism of "We'd sing and we'd pray" emphasizes the word "we," suggesting that a more welcoming, pleasure-oriented church would invite everyone in—and make everyone want to sing and pray, as if it came naturally to them.
In such a church, he goes on, with cheery anaphora and polysyndeton:
[...] the Parson might preach & drink & sing.
And we'd be as happy as birds in the spring:
And modest dame Lurch, who is always as Church,
Would not have bandy children nor fasting nor birch.
And God like a father rejoicing to see,
The momentum of that string of "and"s shows the speaker caught up in his imagination, carried away from the dreariness of his present circumstance toward a more joyful version of church.