I would guess there is very little chance that registration will re-open. Also, the 5K and 10K sold out in approximately 1-2 hours so if it does open back up for some reason it would likely be a very unpredictable and short window. Your best chance to find entries to the 2016 races would probably be to look through travel providers and charities, but those may also be pretty sparse at this point. I have never registered through a travel provider or charity, so I'm sorry to say I can't really provide any insight into how the process works, which ones are the most reputable, etc. but there are some listed on the registration tab on the rundisney website.
Registration usually opens to the public in late April (there is a limited early registration period for passholders and DVC, I believe). Registration usually sells out very quickly for the 10K, 5K, Half Marathon, and Dopey Challenge (usually in that order), so if you are unable to register for 2016 but still have interest in 2017 it would probably be wise to keep it in the back of your mind to check when registration opens and be ready. I believe there has been a sign-up on the rundisney webpage the last few years allowing you to get e-mail notifications on when race registrations open.
The first race I signed up for was the Half during Marathon Weekend 2012 and I did so in mid-September 2011 after registration had been open for a few months. This wasn't intentional, I never even knew of the existence of Marathon Weekend (or entertained the idea of running even ten steps) until that August. This year if someone waited a few hours after registration had opened it would have been sold out. So the popularity and rapid sellout of these events has really escalated quickly; to the point it has basically become a bit of a mantra that the hardest thing about running at Disney is getting registered.
