Maybe I'm totally confused, but a locally free sheaf over an affine variety should be nothing else than a free module over the ring of global functions.
Push forward along the projection is then just restriction of scalars along the inclusion
$$O_X\rightarrow O_X\otimes_k O_Y$$
Now $O_Y=\bigoplus k$ so 
$$O_X\otimes_k O_Y=O_X\otimes_k \bigoplus k=\bigoplus O_X$$
so free modules stay free.