This is a question about genetic engineering in humans.
Hypothetically, could a 47th chromosome containing recombinant DNA be
introduced into every cell in the human body via a viral vector or
would a whole chromosome be too large a molecule for that?

For a start, you need to add two copies of what would be a 24th
chromosome. The current 46 are 23 pairs, not 46 unique structures.
It is already possible to create functional artificial chromosomes in
yeast (YACs), so the basic technology is available. The size is up to,
within limits, there is nothing to say it has to be enormous.
Getting them into every cell of an adult is not practical, and a viral
vector is not a likely way to go about it. The way to change the
genetic makeup is to alter the egg or zygote, and let the normal
process of mitosis take care of getting a copy into every cell.

mikewa is correct, I know that plenty of work has been done in this
area, at least with mice.  You can generate a YAC carrying mouse. The
problem with this is a question of utility.  If you were hoping to
correct genetic disorders in a fetus, they would be detected far to
late for this to be of use.  An exception certainly could be IVF.

To answer your "can a whole chromosome fit" question: In trisomy 21
(Down syndrome) an extra chromosome fits into every cell.  But
chromosome 21 is pretty small.

http://www.ds-health.com/trisomy.htm