To make it vegan is simple:
- Replace the milk with any other kind of milk (soy, rice, hemp, etc.)
- Replace butter with some kind of margarine (I recommend Earth Balance, but just pick anything that doesn't have trans fats)
- Replace eggs with either commercial egg replacer, or apple sauce
As for making it moister, I have no idea.
There are lots of people who have a rather simplistic approach to nutrition and think that removing fat and calories makes you healthy. Then they go through recipes for things they want to eat, replace the sources of fat with something which doesn't have fat and doesn't make the result outright inedible, and declare their recipe a success. I think this is what happened here.
In a cake, eggs provide leavening, moisture, smoothness, own flavor, and enhancement of other flavors. Oil provides smoothness and enhancement of other flavors (and possibly its own flavor, if not netural). And while it is not water based, it keeps the moisture in the cake from evaporating, so it makes the cake less dry.
If you are a "simplistic nutritionist" without all this information, you can approximate some of the effects with soda. It will provide moisture, and it will also provide some leavening because it is fizzy. It will provide some flavor of its own too, but frankly, I find the rather chemical flavor of soda to be unpleasant. And it won't have any fat. In the eyes of the simplistic nutritionist, it has successfully replaced the oil and eggs while reducing fat and calories.
From the point of view of a baker, the cake will be a disaster, and won't even deserve the label cake. It will dry out quickly because it has no fat. It will have a bland flavor. Its texture will be terrible. They say "more chewy?" It will miss both the protein structure and the emulsifying agents provided by the eggs. It will be essentially an overwhelmingly sweet quickbread with no redeeming qualities. From a culinary point of view, it will be terrible.
Bottom line: under some assumptions, it is a good substitution. For me, these assumptions are so far from reality as to be useless. It is a terrible substitution.
Best Answer
You don't need to do this conversion. Most pudding mixes don't contain dry milk. And the mixture is in there to provide starch, not anything else.
The first and best way to make a cake is to start with a good existing recipe. Getting a substitution right is not trivial, it requires some theoretical knowledge and a few iterations of making the recipe and comparing it side-by-side. It can taste OK the first time, but a tested recipe will taste great from the first time, without giving you trouble. Moreover, this one has been converted from normal ingredients to substitutions for some reason. Back-converting it is unnecessary.
If you are absolutely decided to go with this recipe and make the substitution, you can use cake flour. Forget the pudding and its extraneous milk and sugar, and calculate the gluten content of your cake if made with cake flour (use roughly 10% for AP flour, 0% for pudding powder and whatever percentage is printed on your cake flour package, typically 6-7%). If replacing all the flour with cake flour does it, that's best. If the ratio is off, calculate how much pure starch to add to AP or cake flour to get the gluten amount right while keeping the total flour amount constant (which is the weight of your flour + the weight of the pudding from the original recipe).
If you think the original pudding made a difference in fat content, just use cream instead of the liquid milk. The more fat and less water you have in a recipe, the more cakelike it becomes, the most common types of cake use no water or dairy at all. Milk makes your cake more quickbread like. And if that's the effect you want, don't bother replacing anything there just because there might or might not have been dry milk in the author's pudding mix.