No, you do not, at least not for informal social media or business cards, because "MyBook" can be your company's trademark. For example, go to Facebook's home page and look at where they use Facebook, Inc. (The "Inc." part is only present in the legal spiel, not in the front page logo, etc.). I'm holding a Facebook business card in front of me and nowhere does it say "Facebook, Inc.". :)
Just make sure you aren't infringing on someone else's trademark - i.e. "MyBook" isn't confusingly similar to someone else's "MyBook" in the same sector/region.
HOWEVER, You DO need to use the formal entity name "MyBook LLC" on any legal or contractual documents (this includes terms of service/use or disclaimer notices).
Disclaimer: This post is not legal advice and does not constitute an attorney-client relationship.
This is dangerous. I have been coached by lawyers and accountant types to be careful to ALWAYS use the "LLC" on the name. The issue is, as explained to me, that the authorities can make a case if you do not use the "LLC", that you are operating the organization as an extension of your personal finances. And if they can make that case, then you can loose the legal protection you get having the organization be an LLC.
By this I mean, if the authorities decide the organization is just another part of your personal activities, then if someone sues the organization for some perceived loss, they can get access to your personal holdings (like bank accounts, house, car, stocks, whatever you have). The "LLC" is there to clearly indicate to everyone who sees it that it is not a part of your personal activities, but is a separate legal entity.
According to this article, you need to file a DBA if you use the name without the LLC.
There is a huge gray area here though. I'm pretty sure you don't need to use the LLC everywhere you use your business name. It might be enough to use the full name with the LLC on the footer of your website and use the name without the LLC elsewhere.
I bet many companies use the name without the LLC and don't file a DBA. For many states, you file DBAs with the county government. Seems anachronistic for a web company to have to do that.