Re: Will silver nitrate react with a stainless steel container?

The answer to your question is a firm "maybe."

Stainless steels come in various alloys.  They largely are resistant to
chemical attack and corrosion because of the formation of a protective
chromium oxide layer.  If that layer is continuous, there is a good chance
that the stainless steel won't react with many chemicals, including silver
nitrate.

When silver nitrate is put in contact with stainless steel, the iron in the
stainless steel may react with the silver nitrate, causing silver to
deposit on the stainless steel surface as iron dissolves into the silver
nitrate solution.  This is a process called "immersion plating."  You can
read more about it here: http://www.finishing.com/faqs/silverathome.shtml

Your first thought might be "great; a silver coating will stop any more
reaction between the silver nitrate and the stainless steel."  The trouble
is that an immersion plate is very thin and tends to be somewhat porous, so
it isn't a good way to try and put a chemically protective layer on top of
the stainless steel.  More than likely, if some immersion plating starts,
it may well continue.

To insure that stainless steel will be less likely to react with silver
nitrate solution, the stainless steel could be passivated.  The passivation
process removes free iron from the surface of the stainless steel, leaving
behind a nice, oxidized surface.  The test to see if stainless steel is
passivated is to apply copper sulfate solution to the surface of the
stainless steel and see if any copper is deposited on the surface of the
steel (by the process of.....immersion plating).  You can read more about
stainless steel passivation here: http://www.iftworldwide.com/white_paper/passivation.pdf

So, in principle, you could use a stainless steel tank if it was fully
passivated.  The problem, though, is that silver has a greater tendency to
deposit on surfaces than copper; so a surface that shows no tendency for
immersion plating with copper sulfate might still start immersion plating
with silver nitrate.  The other problem is that you can damage the oxidized
layer on stainless steel by scratching it, by improperly cleaning it, or by
putting chlorides on the surface of the stainless steel.  Some guy who has
salty french fries for lunch and then handles your stainless steel tank
without washing his hands just might leave behind a fingerprint that may
well be visible due to silver immersion plating.

So, I've tried to give you the pluses and minuses of trying to use
stainless steel for this application.  Stainless steel may work well for a
short term application, but over time may not be the best alternative. 
That's why glass, plastic, or porcelain covered trays are more commonly used.

Thanks for the question and good luck.