Why is my Buckwheat soba noodle dough crumbling?

The crumbly texture is, apparently, quite common when working with an all buckwheat dough since it doesn't develop gluten - one of the reasons why many home-made soba noodle recipes use a percentage of wheat flour to make the dough easier to work with.

If you have your heart set on a buckwheat-only soba noodle, be aware this is a common problem. You might try boiling water and rolling and cutting a little thicker so they don't end up so fragile, mentioned below. I also saw that the brand of buckwheat flour may be important (using a Japanese style, rather than American style buckwheat flour, apparently makes a big difference). Check that the flour type is correct, that may help.

Another recipe I saw "pre-soaked" the flour (making a dough with water and a bit of lemon juice, and letting sit under a wet cloth overnight) to give the dough time to hydrate the flour. I'm not sure whether the lemon juice or the extra long resting time was the relevant factor, but she doesn't seem to have problems with the dough.

And the last suggestion I found involved "beating the buckwheat", that is, mechanically working the dough long enough for the water-soluble parts of the flour to dissolve into the dough and let it become elastic that way (this extra working may be related to the extra-hydrating time of the last hint, or perhaps actual, mechanical thumping might be helpful in working the dough - it is used in some kinds of recipes to make dough behave).

From an article I found, here:

Not only does a dough of 100% buckwheat flour tend to crumble and break while you work it, but it also dries out incredibly quickly and the resulting noodles are very fragile. Glenn Roberts of Anson Mills recommends using boiling water when making soba with 100% buckwheat to gelatinize the buckwheat starches and help the dough hold together. Roll the dough a bit thicker than recommended in the recipe below, and once made, cook and eat the noodles immediately before they dry out.

You should try the pre-gelatinization method whereby you dissolve the salt in boiling water and pour that over your flour and knead well to encourage formation of structure prior to rolling. Should always use light buckwheat flour strictly avoid dark buckwheat or black buckwheat flour as this will only lead to frustration and failure in the dough. Typically you want to add 30% white flour to 70% buckwheat flour to give you some room for mistakes but without compromising the flavour of the noodle.