No, permits are not required. Under Finland's "everyman's rights" (jokamiehenoikeudet, allemansrätten), you can legally camp anywhere -- even private property -- that's not a built-up area, too close to somebody's house or otherwise legally prohibited (military or border zones, some nature reserves, etc): https://www.visitfinland.com/article/everymans-rights/
The one major catch is that lighting a campfire does require permission from the land owner. Specifically, in the case of national parks, lighting open fires is strictly forbidden outside designated areas, and this is why most people camp at designated campgrounds. (Camp stoves are OK.) Some national parks also ban all camping outside designated spots, so check before you go.
Also, Finland's national parks have an extensive network of cabins.
Some are first come, first served, but the more popular ones do require you to book ahead.