Is it possible to tell anything about market structure if I have annual price data of the industries?
That that we have linear supply and demand functions, pretty much the simplest case such that quantity supplied of good $i$ is:
$$ Q_{s,i,t} = \alpha_{0,i,t} + \alpha_{1,i,t} \cdot P_{i,t}$$
and quantity demanded is:
$$ Q_{d,i,t} = \beta_{0,i,t} + \beta_{1,i,t} \cdot P_{i,t}$$

In equilibrium  $Q_{i,t}=Q_{d,i,t}=Q_{s,i,t}$ so substitute and solve:
$$ Q_{i,t} = \alpha_{0,i,t} + \alpha_{1,i,t} \cdot P_{i,t} = \beta_{0,i,t} + \beta_{1,i,t} \cdot P_{i,t}$$
$$ P_{i,t} = \frac{\alpha_{0,i,t}-\beta_{0,i,t}}{\beta_{1,i,t} - \alpha_{1,i,t}}$$

As you can see, the price depends on supply and demand. Unfortunately, shifts to the level or slope of either the supply or demand functions both affect the price. Therefore, in  general, you can't tell anything about the supply function from the behavior of the price. And if you can't tell anything about the supply function I don't see how you'd know anything about the IO of the producer industries.