TL;DR (too long, didn't read): Yes - it is possible. But not in the most ideal way.
There is no out of the box way yet to enable this. This is a Salesforce Idea with only 350 votes. However, as per that link, and this one, and this one here, a solution is to create a custom formula field to achieve this.
Another method would be to enable Social Accounts and Contacts, as per this link here. The downside here though, I believe, is that the logo would only appear whenever one logs into their own social network and searches for the Account. Whenever one is logged out, they would have to re-do that process in order to see the image display.
Furthermore, Social Accounts and Contacts are entirely dependent on the Facebook, Twitter and LinkedIn APIs and are subject to change. For example, LinkedIn used to be supported under Social Accounts and Contacts however, this is no longer supported.