The viral site looks to be set around the time that the film was actually released, with the exception of the urgent letters to stockholders that appear on some pages. The website actually does display a rapid drop in Masrani stock in December, which is further inconsistent with the summer-2015 setting of the rest of the website.
But the film takes place mainly on December 8th and 9th of 2015, as CT-1138 said, based on the phase of the moon shown in the film:
Here's a lunar chart for 2015:
For some reason, the first time I did this, it showed the 18th and 19th as the days that the right half of the moon would be illuminated. Now it shows it as the 8th and 9th. I'm not sure why that is.
EDIT: Found a link here to a site that shows the moon as it was in the film on the 18th/19th, not sure what's going on. But according to this website it could very well be the 18th and 19th.
We know that the film takes place in December due to the Christmas music playing in the introduction and the fact that there's snow on the ground at the Mitchell residence. The only alternative is that in June it randomly snowed a couple inches in Madison, Wisconsin and Dane County Airport started playing Christmas music ironically, which I doubt is the case.
Visit the lovely Second Earth!
The Forgotten World Characters: Cessna (Pteranodon longiceps masranii ♀)