To answer your question we just need to write it as linear algebra equations and solve them. Although your question doesn't state it, I assume that $v$ and $d$ are unit vectors. Let's call the projected point $x$.
First, because the projected point is in the direction $d$, we can write:
$$\vec{vx} = \lVert\vec{vx}\rVert d $$
Second, because $p$ and $x$ are on the plane and $n$ is the plane normal, we can write:
$$\vec{vx}\cdot n = \vec{vp}\cdot n$$
We can now work on the equation until we get a definition for $\vec{vx}$:
$$\begin{align}
\vec{vx}\cdot n & = \vec{vp}\cdot n \\
\lVert\vec{vx}\rVert d\cdot n & = \vec{vp}\cdot n \\
\lVert\vec{vx}\rVert & = \frac{\vec{vp}\cdot n}{d\cdot n} \\
\vec{vx} & = \frac{\vec{vp}\cdot n}{d\cdot n}d \\
\end{align}$$
Or written differently:
$$x = v + \frac{(p-v)\cdot n}{d\cdot n}d$$
The result is undefined when the scalar product $d \cdot v$ is $0$, which happens when $d$ and $n$ are orthogonal, when $d=0$, or when $n=0$. The first case means $d$ is parallel to the plane and the projection doesn't have a point solution, and the two other cases mean that either the plane or the projection are not defined.