Where to point a low-orbit ion cannon (asking for a friend)
Python 3.8, 116 110 bytes

lambda p:[x:=min(p)[0]+.1,sum(sorted(b+(d-b)*(x-a)/(c-a)for(a,b),(c,d)in zip(p,p[-1:]+p)if(a&lt;x)^(c&lt;x))[:2])/2]

Try it online!
Input: A list of points, each is a tuple of 2 integers.
Output: A point as a list of 2 numbers.
Approach
Draw a vertical line through the polygon. Select the first 2 points where the line intersects with the polygon edges. Any point between those 2 points must be inside the polygon.

Note that the line should be chosen such that it doesn't intersect any polygon vertex.
Code explanation
The x-coordinate of the line is selected as the minimum x of all vertices, plus a small amount (0.1 is used in this case):
x:=min(p)[0]+.1

This ensures that the line intersects the polygon, and that it doesn't pass through any vertex (since vertices all have integer coordinates).
For each edge through vertices \$(a,b)\$ and \$(c,d)\$, the line intersects the edge if \$a&lt;x&lt;c\$ or \$a&gt;x&gt;c\$. Golfed using xor trick:
if (a&lt;x)^(b&lt;x)

The y-coordinate of each intersection is calculated using the following formula:
$$\frac{y-b}{x-a}=\frac{d-b}{c-a}$$
or b+(d-b)*(x-a)/(c-a)
The y-coordinate of the final point is calculated as the average of the smallest 2 y-coordinates.