Are these almond cookies likely to be a success if I make them vegan by substituting coconut oil for butter?
Summary
Pure coconut oil is not an adequate replacement for butter in this recipe. However, coconut oil and water appears to work as a substitution when using ratios between 4:1 and a 5:1 coconut oil to water by mass.
Cookies produced with pure coconut oil fall apart when touched. Ratios of 4:1 and 13:2 coconut oil:water worked in my test.
However, the substitutions tested did not produce an exact match for the recipe produced with butter. (h/t rumtscho for catching the omission of reasons to predict this outcome).
Empirical Tests
Test Procedure
I used a batch size that was 1/3 that of the included recipe, or

32g Almond Flour
7g Confectioner's Sugar
Equivalent* of 1/2 ounce butter
A pinch of salt (quantity too small to measure)

The butter equivalents tested were:

15g Coconut Oil
13g Coconut Oil and 2g water
12g Coconut Oil and 3g water
1 tbsp butter, as marked on the butter package (16g in this trial)

Batches were mixed in the order specified in the butter equivalents list on the same equipment. All batches were placed on a single tray and baked at the same time
Two of the batches were divided into 6 cookies instead of 5 to make them distinguishable from the other batches.
Results
Coconut Oil, No Water
Cookies produced with a 1:1 replacement of butter with coconut oil were extremely fragile. I believe this reproduces the result reported in the complaint cited in the question.
I decided to give up trying to transfer them to a cooling rack after fracturing all cookies in the batch multiple times, so I do not have post-cooling observations for this batch.
Coconut Oil and Water, (13:2) ratio
Cookies hold their shape when handled but crumble readily when chewed. Compared to butter cookies, these are slightly harder along the bottom, and have more snap (it takes some effort to break by bending).
Coconut Oil and Water, (12:3) ratio
I was not able to notice a difference between cookies of this batch and cookies of the 13:2 coconut oil to water ratio batch.
Butter
Note that 16g of butter (instead of the planned 14g) was used, possibly skewing comparisons.
Cookies produced with butter produced results that were between that of the coconut cookies with and without added water; they crumble easily in the mouth but not during handling, and can be broken easily by hand.
Conjecture and References
Some wheat shortbreads are made with oil and water instead of butter, so having an oil and water emulsion as a replacement for butter may not be critical; therefore, it may suffice to simply replicate the fat:water ratio of butter, which is approximately 5:1 by mass.
The existence of vegan shortbread recipes that use vegan butter substitutes suggests that commonly-used vegan butter substitutes may also be acceptable; building on the earlier conjecture, it may suffice to replicate the fat:water ratio of a commercially-available vegan butter substitute. Earth Balance has a 4:1 fat:water ratio by mass.
The original conjecture before empirical tests was that coconut oil and water added in a 5:1 or 4:1 ratio by mass would be an acceptable substitute for butter.
Dairy Butter Characteristics
Dairy butter has a nonzero amount of water. This American source reports about 16% water and 82% fat for American butters.
Using these numbers, this recipe contains about 42.5g butter, which contains about 6.8g (or ml) water. Substituting butter with pure oil removes 6.8 out of 9.3ml of non-oil wet ingredients (2.5ml from the vanilla extract); if non-oil wet ingredients are important, this is a significant change.
As noted by rumtscho in a comment, butter is an emulsion of water and fat and may behave differently in a recipe compared to adding an equivalent amount of fat and water. A difference between butter and coconut oil with added water was observed in empirical tests.
Vegan Butter Substitutes
Earth Balance original spread (easier to read form here) has a significant amount of water. From the calorie counts (assuming 9 Calories per gram of fat) and the fact that ingredients other than oil and water are listed as  &lt;2% ingredients, Earth Balance appears to be around 20% water.
Other Shortbread Recipes
So far, I have not found a directly-comparable recipe.
I found three recipes for wheat shortbread; one without any added water, another that adds water in the form of syrup, and a third that suggests adding plant milk as a binder but discourages it. The first recipe is evidence against the inference that added water is strictly necessary. The third recipe is weak evidence that it is not necessary to have all fat and water emulsified beforehand.
However, almonds and wheat flour have considerably different characteristics, so it may not be straightforward to generalize from successful wheat flour recipes.