Some festivals do, some don't. Some have awards strictly dedicated to social, there also some that take a web series full branding/presence into consideration. It really does vary depending on what specifically is being asked of. That in mind, I'd say to cover your bases in that sense.
From a press stand point, I created and run netTVnow, a website dedicated solely to web series. As someone who receives numerous emails a day from creators inquiring on coverage, it is so vital that I see some sort of social presence. Whether it be a series' actual pages or even just the creators personal portfolio etc. It's very rare that I'm approached by a web series with zero social media presence. At the very least, each series has had at least one social media platform. And speaking for web series that have gotten picked up by distributors, they want series that, yes, have a great story/production value, but also a story that has an audience. Which leads to the question, how do you build that audience?
You'll learn that the web series community is very collaborative, networking is key. Once you get into the community a bit more, you'll find that your stuff gets shared etc. but without a social platform, there isn't really much room for you to grow.
You want to be able to have a place people can engage with the series. It makes all the difference.