The attached patch should accept newlines.  It breaks with two
conveniences though:

1. Field boundaries with character fields are gone.  I had to remove
that step since it is done once only (at widget-setup time) and not
repeated in the after-change hook.  If I had left that in, character
widgets would appear visually different with regard to whether their
initial value was a newline or not.  In the first case there would
_always_ have been an empty line after the line containing the field
regardless of whether the modified value is a newline or not.  In the
second case the empty line would appear iff the modified value is a
newline.

2. The size of a character field has changed from 1 to 0.  Hence if you
erase all characters in the field, there won't be any visible clues left
that there's a field here.  I had to change that since otherwise
wid-edit would have automatically inserted a space (in its after-change
hook) after a user erased all characters in the field.  That space could
have been subsequently used by `custom-variable-set' instead of raising
an error.

Note that tabbing doesn't stop at empty widgets.  I don't consider that
too serious since empty character widgets are invalid anyway.

I tried to keep any side-effects of the patch within the character
widget environment.  Nevertheless you should test it extensively.