Cast to boolean, for integer linear programming
I think I can do it with one extra binary variable $\delta \in \{0,1\}$:

$$
-100y \le x \le 100 y 
$$
$$
 0.001y-100.001\delta \le x \le -0.001y+100.001 (1-\delta) 
$$

Update

This assumes $x$ is a continuous variable. If we restrict $x$ to be integer valued, then the second constraint can be simplified to:
$$
 y-101\delta \le x \le -y+101 (1-\delta) 
$$