Does printed fabric from spoonflower bleed or stain other fabrics if it is stored with it?

That looks like fantastic fabric. Very high quality. 

They have a customer service number, call and ask them. How do they recommend storing the fabric. Will it bleed while storing  it or washing it.

I've got yards and yards and yards of printed fabric, stored for a long time. I have yet to see any fabric bleed colour into any other.  If the fabric gets damp while being stored, if it's poor quality, and dyed in certain colours that tend to bleed such as red and teal, I suppose it's possible but it's not likely.

A great many fabrics can transfer dye when stored against other fabrics. In my experience, the most likely culprits are printed woven cottons.

Usually normal laundering before storage stops this. (That's what I do.) Plus, you always want to wash fabric before you cut it, since it could shrink or twist. Better to let it do that before you cut it to size.

People more dedicated to order than I am launder it, dry it, fold it neatly, and put each in its own plastic bag, where it can't possibly affect whatever fabrics would otherwise touch it.

FWIW, I have a vast fabric stash and I sort first by type (i.e., woven cottons, jersey knits, fleece, ITY, etc.) then by color, so that red cotton print is never up against anything light in color.