Learn Portuguese (Portugal): Ela não vai e eu também não.

In Portuguese, there isn't a single exact word for "neither" in this context. To agree with a negative statement (like saying "neither am I" or "me neither"), you use também não, which literally means "also not".

Why is there no verb at the end of the sentence?

Just like in English, you don't have to repeat the verb if it is already obvious. The full thought is e eu também não vou (and I am also not going), but it is very common and natural to drop the verb vou at the end.

The English translation says "is not going", so why is there no word for "is" before vai?

The verb ir (to go) is doing all the work here. In Portuguese, the simple present tense often covers both general facts ("she goes") and actions happening right now ("she is going"). So, ela não vai means "she is not going" all by itself.

Could I say também não eu to match the English word order of "neither am I"?

No, that sounds unnatural in Portuguese. When giving a short "me neither" type of response, the subject pronoun (eu) goes first, followed by the negative phrase: eu também não (literally "I also not").