1. Applying the Pigeonhole Principle:
Since there are 100 vertices and 10 colors, by the Pigeonhole Principle, there must be at least one color that is used for at least $\left\lceil \frac{100}{10} \right\rceil = 10$ vertices. Let's denote this color as color $C_1$.

2. **Considering the 10 vertices of color $C_1$:**
We have 10 vertices all colored with $C_1$. We need to check if any of these vertices are diametrically opposite to each other.

3. Checking for diametrically opposite vertices:
If there exists a pair of diametrically opposite vertices among these 10 vertices, let's denote them as $A$ and $C$. Since the 100-gon is regular, the vertex diametrically opposite to any vertex $A$ is exactly 50 positions away from $A$. Thus, $C$ is 50 positions away from $A$.

4. **Forming a rectangle with another vertex of color $C_1$:**
If we can find another vertex $B$ of color $C_1$, then the vertex diametrically opposite to $B$ (denote it as $D$) will also be 50 positions away from $B$. The vertices $A, B, C, D$ will form a rectangle, and since $A, B,$ and $C$ are of the same color $C_1$, the rectangle $ABCD$ will have vertices of at most 2 colors.

5. Considering the case where no diametrically opposite vertices are of the same color:
If no pair of the 10 vertices of color $C_1$ are diametrically opposite, then each of the 10 vertices of color $C_1$ must have their diametrically opposite vertices colored with one of the remaining 9 colors. This gives us 10 vertices (each diametrically opposite to one of the 10 vertices of color $C_1$) and 9 colors.

6. Applying the Pigeonhole Principle again:
By the Pigeonhole Principle, among these 10 vertices, at least two must share the same color. Let's denote these two vertices as $E$ and $F$, and their diametrically opposite vertices as $G$ and $H$, respectively. Since $E$ and $F$ are of the same color, and $G$ and $H$ are of color $C_1$, the vertices $E, F, G, H$ form a rectangle with vertices of at most 2 colors.

Thus, in both cases, we have shown that there exist 4 vertices of the 100-gon which form a rectangle and are colored in at most 2 colors.

$\blacksquare$