One easy way is to find the bounding box for your polygon and use rejection sampling: sample from the bounding box and accept if it falls within the polygon, which will happen with probability $1/2$ at least (I think).
Another possibility is to triangulate your polygon. First sample a triangle in a proportionate way, then sample a random point in the triangle. The latter is simple: up to affine transformations, all triangles are of the form $\{(x,y) : x,y \geq 0, x+y \leq 1\}$. To sample uniformly a point from that distribution, first sample $x \in [0,1]$ according to the density $2(1-x)$ (i.e. sample a uniform $r \in [0,1]$ and compute $x = 1-\sqrt{1-r}$) and then sample $y \in [0,1-x]$ uniformly (i.e. sample a uniform $s \in [0,1]$ and compute $y = (1-x)s$). An even simpler method is to sample $x,y \in [0,1]$, and if $x+y > 1$ replace $(x,y)$ with $(1-x,1-y)$.