Short version: no. Slightly longer version: Don't be silly, of course not.
Let's start with what signing a contract actually means. Signing a contract means that you, in your capacity as a legal entity, are capable of understanding the contract, are making an intentional choice and have agreed to abide by the terms of the contract and any obligations or protections that it specifies.
In Second Life an avatar is actually only a placeholder representation of two rather more subtle concepts. All in all, there's an account, an agent, and an avatar. The avatar is the bit that you actually see. The agent is the bit that moves, talks and takes actions. The account mediates permissions.
You log into your account, your agent is placed at a location in-world, and your avatar is drawn in the vicinity of your agent. Usually the agent is an invisible point near your avatar's crotch, but the avatar can be drawn up to ten metres away from the agent.
When you choose to take an action the account is tested to see if the action is permissible, the action is then performed through the agent, and lastly there may be some side-effects on the avatar to represent visually what action is taking place.
None of these three items is a legal entity. None of them are capable of understanding the contract. None of them are able to make an intentional choice.
Anne Loucks made a device so that her cat could make the necessary key-press to click through EULAs (End User License Agreements). Did the cat agree? No. Did Ms Loucks agree? Yes.
Let's compare this with a more practical example away from the keyboard. I am handed a contract. I pull out my pen and sign it. Did my pen agree to the contract, leaving me free of obligation? No. The pen is an instrument (a proxy, if you will) with which I am causing agreement to be signified. The pen can't agree to a contract, so I must have to have done so.
I can't reasonably claim the pen is a party to the contract. Ms Loucks cannot reasonably claim that the cat is either, as she engineered the means by which agreement was given. Likewise, an avatar can't be a party to a contract any more than a pen, a cat, a sock-puppet or a houseplant.
Any reasonable court would immediately identify me, Ms Loucks or the user of an avatar as the responsible party -- because there is no other party that could be.
Now I'm going to say something just a little shocking. You aren't your avatar. Let's pause a moment for the obligatory, involuntary gasp of horror and perplexity.
Your avatar is a digital creation that shadows your actions within a virtual environment and visually represents them. Those visual representations aren't even entirely under your control. Assorted bits of software cause the avatar to do some things that are independent of your actions too.
Your avatar can't agree with anything. Walk away from the keyboard and come back when your avatar agrees with something. Better yet, read the rest of this blog, or maybe the comics. Because you'll have a whole lot of time to kill. Your avatar can't think or act or understand or agree. You are the one who does all of that.
Now, does that mean that Tateru Nino can't agree to a contract? No. Because that's me, and I'm a legal entity and all that. Tateru Nino is a nom de plume which is also attached to my account, agent and avatar. I can (and do) sign contracts with that name. It goes on my tax forms. Did Reginald Dwight sign his contracts as Elton John or as Reginald Dwight? (Answer: as Elton John).
It is legal to do in most jurisdictions, so long as you demonstrably have "no intent to deceive". If you're using an alternative name to avoid your obligations, then that's pretty much going to add fraud to any penalties for breach-of-contract.
How is it that Tateru Nino can agree to a contract (in Second Life or not), but the Tateru Nino avatar cannot? Because I'm not my avatar either. It's my faithful, and tastefully-dressed servant. My sock-puppet of digital pixels.
And sock-puppets can't agree to contracts either.