Does $E_{TM}$ accpets the empty word $\varepsilon$?
I would answer that no, $\varepsilon \notin L$.

The reason is that the typical definition of a Turing machine is that its representation should contain a non-empty set of states and a non-empty tape alphabet.

In order to represent those, you typically would need a non-empty string.

However, as Yuval points out, you are free to define your own Turing Machine description language, and if you were to say "We use $\varepsilon$ to denote the "empty" TM (with alphabet 0), the TM that runs forever", you are free to do so, in which the answer becomes Yes.

I would go with No, $\varepsilon \notin E_{TM}$.