That box did not look like anything that would hold a piano or related
roll, and what would "web" refer to in that context?

My guess is it held a part for a web press.  I am not very
knowledgeable about this field, as my guess may prove.  But web presses
are a kind of offset printing press that utilize cylinders and rollers
to do their thing.  Perhaps a web roll(er) was a carefully machined
piece of metal that had to be perfectly round and balanced, and 
protected from dings and scratches while in shipment or storage?

Paul Eggert
Fountain Hills, AZ