I will give you a slightly different piece of advice from those above.
However, my opinion is based on your statement that you can, without rushing/cutting corners, get your rifle to an "in the white state" before hunting season.
If that is the case, there is nothing preventing you from using it "in the white" if it is otherwise finished and functional.
I have a 50 cal Ohio that is "still in the white" almost 4 years after it got to that stage. New builds, other hobbies, work and life have all got in the way of staining/sealing the stock (the metal will be left bright as was typical with many Ohio rifles) - and yes, this rifle has been used off and on over those years - including a few days of hunting in my late blackpowder season (December).
So, the short answer is "yes", an "in the white" gun can be used - it will "function" just fine.
I am not sure what extra measures I may have to do/take when (or maybe if) I do get around to finishing it - there is now skin oil, dirt, pollution etc ground into the wood and the wood is starting to take on a more aged look.
But to be fair, perhaps my perspective is different. If this is/will be the only gun you build for yourself (or at least until you are advanced in age, the kids have left and you get bored with the honey do lists), then maybe you should wait and get her done, even if that means she isn't ready until "next season".
I am not a prolific builder but can crank out 3 or 4 rifles a year without working too hard at it - and even now I have two others "getting there" in terms of being finished after having finished two this year already. So, I only "have to" use an in the white rifle if I want to - I have other options.