Kate,  I am not familiar with the "Mills Automatic Phonograph" design,
but do own a 1938 Rock-Ola jukebox which must be from the same era,
having the great wood case, playing 16 records, each on their separate
platform.  In fact, this is the only "electric" phonograph I own, and
it's very much a favorite!

My first advice is simple: *Don't plug it in* until it has been
verified that the amplifier and mechanism is in operable condition.
A common result of aging in older tube-type amplifiers is that the
electrolytic capacitors dry out, and can cause current to either
short out, or be unfiltered/unregulated.  All this means that there
is a potential [pun intended] for frying what can be restored.  Some
restorers have a special setup to enable testing in a limited-current
environment, but I'd test everything I could with a meter even before
that.

Similarly, the mechanism should be tested to see that all works freely
and is lubricated appropriately; overheating a motor or twisting a
frozen joint would be a big disappointment at this time!

I'd suggest you strike up an acquaintance with those who collect
old radios.  The best source for me was the newsgroup
"rec.antiques.radio+phono", although I haven't visited there for a
couple of years.  Someone there should also be able to help you with
supplying tubes, if you really do need them, although "Needs tubes"
is probably just another way of saying "Needs work" without really
knowing anything about the real condition.

Good Luck!

Todd Augsburger - Roller Organs
http://www.bright.net/~allmax/index.htm