Suppose such a 2023-gon exists.
Let its side be denoted by $a$, so $a^2$ is an integer, and its vertices as $(x_1, y_1)$, $(x_2, y_2)$, ..., $(x_{2023}, y_{2023})$. Consider the 2023-gon with the smallest value of $a^2$. We have $(x_i - x_{i+1})^2 + (y_i - y_{i+1})^2 = a^2$ for each $i$, where $x_{2024} = x_1$, $y_{2024} = y_1$.
If $a^2$ is a multiple of 4, then since if the sum of two squares of integers is a multiple of 4, then both numbers are even, we have $x_i \equiv x_{i+1} \pmod{2}$, $y_i \equiv y_{i+1} \pmod{2}$ for each $i$. But then we can consider a polygon with half the number of vertices with vertices at $(\frac{x_i-x_1}{2}, \frac{y_i-y_1}{2})$, whose vertices are also all integer points, and whose side length is $\frac{a}{2}$, obtaining a contradiction.

If $a^2 \equiv 2 \pmod 4$, then $x_i$ and $x_{i+1}$ have different parity for each $i$, which leads to a contradiction, since we obtain that $x_1$ and $x_1$ have different parity.
If $a^2 \equiv 1 \pmod 2$, then $x_i + y_i$ and $x_{i+1} + y_{i+1}$ have different parity for each $i$, which leads to a contradiction, since we obtain that $x_1 + y_1$ and $x_1 + y_1$ have different parity.