It is always difficult to find St. Patrick's Day stories that are appropriate for 2-year olds. I was excited when we just received our copies of There Was an Old Lady Who Swallowed a Clover by Lucille Colandro, because it is perfect for this age group. Plus, it lends itself really well to flannel!
At this time I don't have an old lady, but I may add one made out of paper bag (so the pieces can go inside as she swallows them). You can see there are pieces for the clover, daisy, butterfly, bird, pot, gold, fiddle, leprechaun, and rainbow. If you haven't read Colando's stories before, definitely pick them up. They are perfect story time reads.