Is it indeed possible
and reasonable to craft a carriage side of such large single sheet
It can be reasonable, but typically isn't. To form an entire side of the carriage from one sheet of steel would require a forming press that is gargantuan. However, one could instead have three more reasonably sized presses and then weld the sections together. Further, this could be more flexible - changing out the press plates is easier and faster, storing smaller press plates is cheaper and easier, if the machine breaks down you can still use the other presses, whereas if one piece of the gargantuan press breaks down, the entire operation might be waiting for the repair.
There are a whole host of issues that follow the same reasoning - if a panel is damaged, it's expensive and time consuming to rework, and affects a much larger part of the carriage than a single damaged panel would. The pieces are much harder to work with, move, and fasten, etc, etc.
So it's very unlikely that it's a single seamless sheet of steel or aluminum.
or should I expect that there're neatly hidden seams somewhere?
You should expect seams, and in fact, as ratchet freak pointed out:
"The bodyshell is [...]of full monocoque construction with an all-welded mild steel stressed skin,"
Monocoque is a technique where the skin or surface of the construction serves an integral support purpose.
The seams, therefore, may also provide support as ribs, and so you may find that having seams provides an added benefit, or perhaps reduces the need for internal supports.
The seams, of course, are not visible for both aerodynamic and visual appeal purposes, and it's relatively easy to hide weld lines with grinding, sanding, and polishing.
So without actual proof of manufacturing process, I believe you can safely assume smaller sheets with invisible seams are more likely than single seamless formed sheets.