Fortunately, Nintendo is selling replacement Labo parts. As previously stated in Nintendo's Labo FAQ, you'll be able to get them by going to Nintendo's online store or by calling the company at 1-800-255-3700.
As of this writing, Labo parts aren't listed on the main page, but I found some by typing "Labo" into the search bar. The projects in the variety kit, including the piano ($11.99), motorbike ($11.99), house ($5.99), fishing rod ($8.99) and RC Car and Discover combo pack ($2.99) are all sold in their entirety, though reflective stickers are another $2.99 and an accessory pack with strings, rubber bands and more runs you $9.99.
Of course, Nintendo isn't selling the software separately, so you can't go and buy just one variety pack item and play with it on your Switch.
The robot is far more complicated, because Nintendo sells replacement parts piecemeal. There's the main body ($13.99), knobs ($5.99), sliders ($9.99), straps ($5.99), weights ($7.99) and a combination pack with visor, feet and Joy-Con holder ($5.99). The reflective stickers are $1.99 and the accessory pack with all of the extras is $9.99.
At some point in the future, I fully expect a Labo DIY community to pop up and perhaps even offer stencils to trace on your own cardboard. But if you want the official parts from Nintendo, they are available, and that's good to know. Because cardboard is destined to break eventually.