Determine price and cost of intermediate good
One way to close the model is through the labor market equilibrium. The OP clarified that the final-goods sector does not use labor.  

Let $D_y = D_y(q)$ be the total demand for the intermediate good, in quantities. Since the market for the intermediate good is competitive it will clear, so we have

$$D(q) = AL \implies L^d = D(q)/A$$

Let $L^s(w)$ be labor supply coming from the utility maximization problem of the households and depending also on the wage. Assume a competitive labor market. Then the wage will adjust to clear it. Here we have

$$L^s(w) = L^d = D(q)/A$$

which gives us a second relation between $w$ and $q$, alongside $w=qA$, and it permits to determine both simultaneously.