How does water move through and is stored by vegetables in general and
corn specfically?

Water moves through vegetables through the process called osmosis. 
Corn behaves much like every other vegetable, and water enters and
exits the individual cells through the cell wall and cell membrane.

The amount of water in the respective cell depends on a number of
factors, but among them are what specific type of cell it is (skin,
husk, kernel, etc), and the local conditions.

As with all things in nature, water goes from areas of high
concentration to low concentration.  So, this means that water will
travel across the membrane depending on which side has more water.  It
will always try to balance things out.

You can perform an experiment yourself at home where you have three
tubs of water with varying concentrations of salt.  If you place a
french-fry shaped potato piece in each, you will see the one with the
lowest concentration of salt (highest concentration of water) will be
the hardest, while the other two will be noticeably softer.  As water
exits the potato (in order to balance out the container), you lose
turgid pressure, which is much like hydraullic pressure, so the potato
softens up.

Inside the cell, the chromosomes tell the cell parts like
motichondria, vacuoles, etc. how to behave, and they regulate the
amount of water in the respective cell.  For instance, you may notice
that the kernel appears to have more water than the husk.  All cells
contain roughly the same amount of water, which amounts to right
around 70-80%, but local conditions may change that.  A husk in the
sun will lose its water to evaporation, while the protected inside
will retain its water, for instance.

Water first enters the plant through specialized structures called
root hairs that extend into the soil to increase their contact with
water and nutrients. As touf-ga points out because plant cells are
hypertonic (have higher salt levels) compared to the water in the soil
water enters the root hairs by osmosis. Once in the root hair, which
is part of the epidermis water is passed from cells to cell through
plasmodesmata (plant specific structures that connect nearly all cell
in the plant) eventually arriving in the vasculature. In the
vasculature water is drawn up throughout the whole plant via
specialized cells in its vascular system called xylem. These cells are
very rigid and mostly empty making them good conduits for water flow.
In general you can think of xylem almost like a straw. Water being
lost from the leaves through evaporation/transpiration helps to draw
up water through the vasculature. Of course waters adhesive and
cohesive properties also play a role in this process.

Water must also be able to exit the xylem and pass into other cells.
As touf-ga ponts out osmosis is one way in which this can occur,
however the cell membranes of many cells are very hydrophobic and will
not readily allow water to pass through them. So cells (this includes
humans cells) have other ways to let water through. One of these ways
is by positioning proteins in their membranes that allow specifically
for water uptake. These are called aquaporins. They are essentially
channels for water that also allow for uptake and loss of water. These
also function in the uptake of water initially by the root.