Let's write out the multiplication, so that it becomes easier to see.
$$(1+2x-x^2)(1+2x-x^2)(1+2x-x^2)(1+2x-x^2)$$
We can now see that the only way to get an $x^7$ is by taking three $-x^2$ and one $2x$. There are $\binom{4}{1}=4$ ways to pick which term the $2x$ comes from, and the coefficient of each one is $(-1)^3(2)=-2$. Therefore, the coefficient of $x^7$ is $(4)(-2)=-8, \boxed{\text{A}}$.