In my earlier comments I mentioned

>Typically the pronoun *they*, in this instance, would be italicised, to show emphasis in speech. Sometimes scare quotes are used instead. In the actual episode, did the speaker use [air quotes][1] by any chance? 

When the OP confirmed that italics were not used, and the actor did not do air quotes, I replied:

>No, I wouldn't expect subtitles to go into the trouble of using scare quotes or italics. I suspect Speaker B stressed *they* and gave a quizzical look

The Oxford Dictionary provides the following definition
>*[scare quotes][2]*     
>Quotation marks placed round a word or phrase to draw attention to an unusual or arguably inaccurate use.

So, if we look at the first line cited by the OP we see that the name of an organisation is used, **Hessington Oil** (I've capitalised *Oil* because it is part of a proper noun). In American English — and *Suits* is an American TV series — organisations and companies are often used in the singular. For example,


-  Microsoft **is** one of the industry leaders in accessibility innovation...
- The Coca-Cola Company **has** on occasion introduced other cola drinks under the Coke name. 

It could be that speaker B believes that "they" refers to a company or an organisation, and as a result uses the singular verb in: “Who ***is*** they?” Who "they" is, might also refer to a person, male or female, in which case the verb is singular.  

>**Who** is they?


It's also worth noting that the subject in the question is *who*, which is typically used for people and organisations, not for words. If the speaker had wanted to know about a word they could have asked: “What do you mean by *they*?”  

>Now **they've** decided to take on Ava Hessington personally

Until we learned the answer, it was unclear whether the subject was plural or singular. The singular *they* is a prominent feature in the English language, it is used in place of *he/she* and it is frequently heard in speech. 



  [1]: http://dictionary.cambridge.org/dictionary/english/air-quotes
  [2]: https://en.oxforddictionaries.com/definition/scare_quotes