Freezer door wiring on my Whirlpool side-by-side - can I splice it or do I need a whole new door?

You can technically do it with the door hung, but I'd pull it off. The wire bundle exits through a small channel at the bottom hinge, and you'll need enough room to strip, crimp, and slide heat shrink on. Trying to solder or crimp wires while they're hanging in a tight gap is miserable. On mine I just removed the top hinge cover, a couple of screws, and tilted the door off the bottom pin, then laid it on a towel across two sawhorses. Tape the freezer side shut so the door gasket doesn't get fouled while you work.

For wire, don't grab solid household Romex. You want stranded wire of the same gauge as what's in the bundle, which is usually 18 gauge for dispenser circuits. Most hardware stores sell stranded primary wire by the foot. Better yet, buy butt splice connectors that come with adhesive-lined heat shrink already on them. Strip, crimp, hit it with a heat gun until the ends ooze a little sealant, done. Solid wire work-hardens and cracks at the bend point, which is exactly the failure you're repairing.

A few gotchas. First, check every wire in that bundle, not just the obvious breaks. These harnesses fail one conductor at a time, and the wire that looks fine can have a crack under the insulation a quarter inch away from the visible one. A cheap multimeter set to continuity with the wires flexed will catch the intermittent ones. Second, don't just splice the minimum length. Give yourself a good foot of extra wire so you can route a gentle loop. Third, anchor the splice bundle away from the hinge so the flexing happens in the middle of the soft wire, not at your crimps. Zip ties to an existing screw or bracket work fine.

One more thing before you spend a dime: check your purchase date against Whirlpool's service bulletin on these door harnesses. Depending on age, they may cover the parts, which is a phone call worth making even if you still do the labor yourself.

One thing I'd add from doing this same job on my old fridge a few winters back: photograph the wiring layout and label each conductor with masking tape before you cut anything, because once you're staring at four or five identical-looking wires hanging out of a jacket you will absolutely forget wich went where. I also found the little connector plug where the harness meets the cabinet had some corrosion on one pin, so clean those terminals while you're in there. And mark which wire had the break with a bit of tape before testing, since continuity checks on a jumbled bundle get confusing fast.

Agreed with the labeling advice above, that saved me on my Kitchenaid door harness. One thing I learned after the splice was done and everything looked good: the dispenser still wouldn't shoot water, and it turned out the micro switch inside the door itself had gone bad from all the flexing over the years. So if you get it all buttoned up and the water still won't flow, don't rip your splice apart right away. Pop the dispenser cover and check that switch with a meter before assuming your crimps failed. Also ran the new loop of wire through a short piece of split loom tubing I had lying around, cheap insurance against the hinge edge.