I've started a new game after installing patch 1.3 and am now in Flotsam. I tried to use the Inn storage to store some stuff but every time I choose the storage option in the dialog with the Innkeeper, it just opens the shop.

Had the PanelTweaks mod when I installed patch1.3 but I uninstalled it just after starting the prologue when I heard it caused some problems. Other than that, no mods or anything. Is it just me? Anybody else encountered this problem and found a solution? I'd really like to use that bloody storage.

Alternative would be to install a mod that removes the weight of crafting components, because I'm getting really tired of hunting all over for those whenever I want to get something crafted, and without the storage, I can't really hoard them.

Exactly, the chest is there, you can access it directly. Works fine for me (altough the label button for x360pad is incorrect, but that has no bearing on anything).

The panel tweaks mod is now updated for 1.3: http://www.witchernexus.com/downloads/file.php?id=87

These are a couple of weight mods: http://www.witchernexus.com/downloads/search.php.
I made my own mod, just halved the weight of all items, if you'd like that.

Thank you, I hadn't noticed the chest at all. I should be fine with the storage, but I'll keep it in mind in case I get tired of running back the the Inn all the time.

I also started a fresh install and game when 1.3 came out but the storage chest is not in the Inn. I've looked next to the bar but no luck. Any ideas?

Going to assume since it wasn't mentioned in this thread that you're looking at the wrong bar. The one with the storage chest to the side of it is downstairs in the room with the fistfights.