Short of a divine intervention, there's nothing that can stop Donald Trump's inauguration at this point. The general election, recount efforts, and Electoral College vote all failed, and now there's no last hope to prevent Trump from assuming the presidency. Mother Nature might still be able to have something of a say in the form of a winter storm, but all that will happen if it snows on Inauguration Day is a change in venue.
Unless there's a bonafide blizzard on January 20, it's very likely that the show will go on just as regularly planned. According to USA Today, JFK had the snowiest inauguration on record when eight inches fell the night before he was inaugurated, and the event proceeded without much of a hiccup. However, it wasn't snowing at the time of the ceremony, which would have significantly complicated matters. If it does snow at noon on the 20th, Trump may have to move his swearing-in inside, like Ronald Reagan did in 1985 when the windchill in Washington was a balmy -22 degrees. If his inauguration more resembles JFK's, Trump will be able to continue the inauguration and subsequent parade down Pennsylvania Avenue without any major problems.
While the inauguration would probably still go on in the event of snow either before or during the ceremony, there would likely be a lot less people in attendance. D.C. is expecting over one million visitors for the historic event, and many, if not most, of them will be traveling from out of the area. Highways and airports could be closed if it's a big enough storm, meaning thin crowds out on the National Mall for the ceremony. It could also impact the few artists who have agreed to perform at the inauguration, who might also get stranded by the weather in that case. That won't directly effect the presidential installation, but it would be an interesting karmic turn of events.
Ooh, girl: Idina Menzel says "karma" is behind Donald Trump’s rumored Inauguration Day problems https://t.co/aR4dafqorZ— huffpostqueer (@huffpostqueer) December 16, 2016
The inauguration has already been plagued by disorganization and controversy, so a snowstorm would fit right into that theme. Unfortunately, the weather isn't animate and can't consciously decide to mess with Trump's inauguration. Maybe if everyone comes together and used collective thought to bring about a blizzard. Knowing Trump, however, he'd probably take it as evidence that climate change doesn't exist and the EPA should be eliminated. Either way, snow, no matter the amount, won't change much about Inauguration Day.