While Valheim doesn't officially support mods, modders aren't letting that slow them down. There's already a mod that overhauls the construction system, and so it was probably only a matter of time until someone overhauled your Viking's perspective. Yep, there's now a mod for Valheim that lets you play in first-person mode.
And the while the First Person View mod is still a work in progress, and the game itself doesn't seem like it'd be a natural fit, it really does work surprisingly well. I ran around for about a half hour in first-person this morning, and the beautiful world looks even better when you peer through your character's eyes instead of looking over their shoulders. Here's a little montage:
Combat is definitely a bit tricky—blocking with a shield gives you a nice look at the inside of your shield and nothing else, and it's hard to see yourself actually swinging your weapon. In general there's a bit of clipping through your own body when moving and looking around, but sailing in first-person is definitely pretty sweet.
The really nice thing about the mod is that you're not locked into first-person mode. With the mod installed, the scroll wheel lets you zoom in and out from third-person mode right into first-person. So if you're having difficulty, you can zip back into the vanilla game's POV very easily.
The First Person View mod is dependent on another mod called BepInExPack Valheim, but don't let that dissuade you because it's an easy install. Once you've installed it and run the game once, it'll create a plugin folder that the First Person View mod can simply be dropped in. There are readme files for both mods that explain exactly how to get the mods running, and the First Person View mod also includes a new config file so you can edit values for FOV in both first- and third-person and the font size of damage notifications. You can find it here at Nexus Mods.