The book that answers this question (and indeed should be a 'must
read' for all enthusiasts) is Craig Roell's "The Piano in America",
(University of North Carolina Press, 1989) which looks at the piano
trade from 1890 to 1940.  It covers the player piano thoroughly --
as you would hope it to, but something that most piano history books
omit or underplay substantially.  The piano trade kept sales figures,
which were regularly reported in Music Trade Review, so the basis
for this analysis is fairly solid.

The specific question regarding the proportion in total of player
pianos in homes, as opposed to annual sales, is something that
I strongly suspect there are no direct measurements of.  (How stupid
of the census not to include a question about it!)  Something can
be inferred from annual sales and assumptions about the lifetime of
pianos.  Referring to the sales chart given by Roell (Fig. 2, page
216) gives the following --

Roell's figures go back to 1909, at which point players sold about
30,000 out of a total of 380,000 pianos (say about 10%).  The growth
in the proportion of players from that point on was roughly linear up
to 50% in 1919, so we can say it was maybe 25% in total for the decade
1909 onwards out of average total sales of 350,000 annually.

If we do a pure guess for the previous decade, knowing that in 1909
players formed 10%, maybe about 5% of sales were players out of annual
sales of 400,000 -- although piano sales probably rose rapidly in that
decade so the average annual total could well have been lower.

The player formed about 50% of all sales from 1919 to the later 1920s,
about 160,000 annually from total piano sales of 300,000 (broad
averages up to 1925).  Sales of everything dropped rapidly in the 
later 1920s, but players stuck at nearly 50% right up to 1928.

To estimate the cumulative proportion of players we need to assume
how many non-players there were in 1899.  Let's say 4 million (i.e.,
the same in total to that date as were sold in the next decade).
Let's also assume that no pianos were scrapped in this period.  Given
these assumptions the running total at the end of each decade of pianos
sold up to that point, and the proportion of them that were players,
can be calculated as follows:

  1899: 4m pianos in homes, 0% of them players.
  1909: 8m pianos, 2.6% players
  1919: 11.5m pianos, 10.3% players
  1929: 14.5m pianos, 21.6% players

Given that sales dropped uniformly in the 1920s, it's easy to see that
the highest proportion of players in houses would have occurred at the
point when players ceased to form more than 50% of sales, which
inferred from Roell's graphs was some point in 1926.

Obviously these are the broadest of estimates (or even a fine piece
of fiction), but the final figure of roughly 1 player to 4 pianos feels
about right, and I guess the range of error is plus or minus a few
percentage points depending on what assumptions are made.  The figures
also refer to total instruments manufactured, some of which would have
been exported.

Julian Dyer