Entry permits are available 5 months prior to the entry date.  I'm not sure what you are referring to with respect to "4-days prior."  You should be able to call the park office in Atikokan to determine availability or you can "go direct" to the Ontario Parks permit office (888-668-7275) for your permit.  That is, of course, if your entry date is BEFORE mid-September (the "required permit" season runs from mid-May to mid-September).  After that period, an entry permit is not required.  You simply pay for camping fees.  If the ranger stations are closed down, you'll need to find a "self-service" pay station (I know you can do that at Atikokan, Nym and Dawson Campground, but I cannot personally confirm this for Beaverhouse).  If a permit is required, your chances are better if your entry date is not on the weekend.

dd

Just re-checked the "permit window."  Looks like it now ends in mid-October (Oct 12th).