Greetings Matthew,  Welcome to a great hobby.  Low vacuum on your
Gem Roller Organ could be from any one of a number of causes.
Generally the best course of action is to do a complete rebuild of
the organ.

While you are recovering the vacuum reservoir you can look for any
cracks in the wood that would leak unwanted air into the reservoir.
Likewise in recovering the pump bellows you may examine for any wood
splits in them.  The valves on the inside and outside of the two pump
bellows should be replaced with new valve leather which will give you
an immediate boost in vacuum as these most certainly have gained in
porosity with age.

Once you do this simple work (using only hot glue, leather, and rubber
cloth) you will be amazed at your instrument.  While apart you can also
clean the reeds very gently with a puff bulb to remove dirt.  I would
not recommend much more than that for a beginner as the reeds are the
only delicate part of the organ.

Some organs were fitted with a limit valve on the vacuum reservoir so
that when it closes nearly all the way a peg, pin, or screw head pushes
the valve open to let air in.  If there is a spill valve this too
should be checked for seating and operation.

Lastly, the leather or felt facing on the key valves may have been
removed, compacted, or fallen off.  If all of the keys leak together
you will not have full vacuum and they can leak a little without
sounding a note.  Check the key valve faces to determine whether they
are making a solid contact and sealing well.

As for your question on doing piecemeal restoration of the organ, if
the leather is at the end of its life anywhere on the instrument you
should assume all of it is gone or about to go.  You could fix one pump
bellows today to make the unit play better, but the reservoir or other
pump bellows would soon follow in failure.  I did a complete rebuild on
my first Concert Roller Organ ($5 from a true junk shop) in 1963 and it
still plays very well.

Best of luck and let the MMD list know how you do,

Al Sefl