Tolkien wrote in a footnote to Letter 153 that Hobbits had no practice of worship or prayer:
There are thus no temples or 'churches' or fanes in this 'world' among 'good' peoples. They had little or no 'religion' in the sense of worship. [...] I do not think Hobbits practised any form of worship or prayer (unless through exceptional contact with Elves).
The Letters of J.R.R. Tolkien 153: To Peter Hastings (draft). September 1954
So there's no institutional religion on the Shire. Of course, not worshiping a god isn't quite the same thing as not believing in that god, and spirituality can exist without religion.
That being said, there's no indication or reference to the rank-and-file hobbits having any knowledge or understanding of the creation myth; since Hobbits are an evolutionary off-shoot of Men, it seems likely that they had some kind of religious knowledge or worship at some point in their history1. However, by the time of the Third Age the Hobbits retained little (if any) of their early lore, so it seems unlikely that they would have retained religious traditions as well:
Of their original home the Hobbits in Bilbo's time preserved no knowledge. A love of learning (other than genealogical lore) was far from general among them, but there remained still a few in the older families who studied their own books, and even gathered reports of old times and distant lands from Elves, Dwarves, and Men. Their own records began only after the settlement of the Shire, and their most ancient legends hardly looked further back than their Wandering Days.
Fellowship of the Ring Prologue 1: "Concerning Hobbits"
It does seem likely that some Hobbits would have learned of the religious traditions of the Elves (we know at least that Bilbo did, since The Silmarillion is in-universe his translation of Elvish history), but it's not clear whether or not they would have perceived it as religious truth, or as mere historical curiosity.
If hobbit-culture has devised any other form of religious belief (animism, for instance), there is no record of it.
1 As the Gondorians do, for instance; consider Faramir's "grace at meat" in The Two Towers:
Before they ate, Faramir and all his men turned and faced west in a moment of silence. Faramir signed to Frodo and Sam that they should do likewise.
'So we always do.' he said, as they sat down: 'we look towards Númenor that was, and beyond to Elvenhome that is, and to that which is beyond Elvenhome and will ever be.
The Two Towers Book IV Chapter 5: "The Window on the West"