It is a good ideas to stimulate users to report more and partially become more constructive for the forum via their reports.

However, this one, if implemented unwell, might cause a new pressure on forum staffs due to spamming reports.
In my opinion, I think that beside rewards (badges as you proposed), forum should have strict rules (including punishments) for bad reports, especially bad reports in-a-row.

One more thing, what happened if someone use hundreds of newbies account, or even brandnew, publishing hundreds or thousands of shitty posts.
After that, they use their main accounts (higher ranks, of course), to report as their contribution and earn badges.

Admin should consider about all those scenarios, both positive, and negative.
Thank you.