Essentially, a site is just something independent that lives on the Web. How do I actually associate that site as being protected under legal liability of my LLC company?
Does it just happen automatically? Or is there some legal paperwork that needs to be signed?
I'm fairly sure there is no global consensus on how this is done. The one time I went to court over something like this, my lawyer actually brought a large set of screendumps to paper as evidence of the other party's websites ownership status.
What I would suggest is:
It happens automatically by virtue of the fact that your LLC owns and controls the website. This is probably documented by the domain registrar which lists the owner of each domain, and could also be proven by asking the web hosting company who controls the site. The important thing is to make sure the domain is registered in your name, that the contracts you sign with the web hosting company and/or internet provider are in the company's name, not yours, and that any servers were purchased by the company, not by you personally.
If these things are not currently owned by the company, you may want to sign a contract of sale so that you have proof that the company has purchased them.
Also, as a general rule, it never hurts to display the name of your company somewhere, like on the bottom of the home page.
(I am not a lawyer in any jurisdiction and this does not constitute legal advice.)
Ultimately the ownership resides with domain name ownership... All the control of a website can be broken down to the level at which the domain name is managed - this is where you define what IP address it points to etc. So while nobody can come along and just take your website, you need to make sure that the LLC owns the domain name. By owning the domain name, you get to point it at your servers that serve your code. So if your LLC owns the domain name, then you're covered. If you personally own the domain name, and not the LLC, then you're going to need to transfer it to the LLC. Even though the LLC is owned by its members, it is a separate legal entity to the individuals that own it - so to be covered by the LLC umbrella it must own the domain name (and preferably pay for the hosting services etc).