  We study vector bundles on the moduli stack of elliptic curves over a local
ring R. If R is a field or a discrete valuation ring of (residue)
characteristic not 2 or 3, all these vector bundles are sums of line bundles.
For R the 3-local integers, we construct higher rank indecomposable vector
bundles and give a classification of vector bundles that are iterated
extensions of line bundles. For R the 2-local integers, we show that there are
even indecomposable vector bundles of arbitrary high rank.
