I am not completely sure what the question is here so let me say something about the picture in general. There is no "vanishing K3" in Vafa's description. The singular K3 geometry that Vafa is talking about is a K3 surface with some ADE singularity. These surfaces are of finite size from the metric point of view - both their volume and their diameter are finite. The only difference between them and their resolutions is that the resolutions have additional Kahler parameters - the volumes of the exceptional spheres of the resolution. In other words, the singular K3 surfaces are parametrized by special loci in the interior of the Kahler moduli - the equations of these loci are given by setting the volumes of the exceptional spheres to zero.

You can also describe these singular K3s as complex (rather than Kahler) degenerations of smooth K3s, and I suspect that this is what you are after with your question. K3 surfaces with ADE singularities appear naturally as special fibers in one parameter families of K3 surfaces where the total space of the family and the general fiber are smooth. The typical picture is that you have a smooth complex threefold $X$ and a proper holomorphic map $f : X \to D$ to a one dimensional complex disk centered at zero, so that the fibers $X_{t} = f^{-1}(t)$ corresponding to $t \neq 0$ are all smooth K3 surfaces, and the special fiber $X_{0} = f^{-1}(0)$ is a K3 with say a single ADE singularity. Are you asking for specific models of this setup. There are many ways to get these, e.g. by looking at double covers of rational surfaces, or by looking at families of elliptic K3s. The easiest example is to look at a pencil of smooth quartics in $\mathbb{P}^{3}$ degenerating to a quartic with an isolated ordinary double point (which is in the ADE nomenclature is just an $A_{1}$ singularity).

Now for the family $f : X \to D$ you can form the nearby and vanishing cycle sheaves at $0 \in D$. They are very easy to describe. For instance the sheaf of vanishing cycles will be supported at the singularity $x \in X_{0}$ of $X_{0}$ and its stalk at $x$ will be the root lattice of the ADE group.