There have been times my mic has been misbehaving and I've just connected using my phone. Works fine, I'm sure you could do that too.
I'd have to say it would be a concern point for me if someone couldn't speak or attempt to speak for an interview, especially if they knew this beforehand. Fair enough if you go into it with everything working and the connection sucks, that's a different matter. But not even attempting to try and answer the questions through a voice channel is a no no from me. It's supposed to feel like a proper interview
Staff meetings are over discord as well so if you want your opinion properly heard, it's better to be able to speak. I often have to type out things because it's usually very late for me when we have meetings, and not everyone sees the messages or by the time you've done, it's the next topic. Stupid timezones haha.
But yeah if you're applying for staff, you need to be on discord and be ready and willing to actually physically speak to people.
~ @EllieEllie
^ This is essentially what I was going to say, but it confirms for me that it's the same as when I was staff.
You need a mic and need to be able to use it to be a staff member. No questions, no exceptions; you need to be able to use it. If connection issues or mic issues occur at some point, that can be worked around, but to apply you need a working mic.
Like Ellie said, it's not just the interview. First of all, if you happen to get staff and moderator, then you'll likely participate in interviews for Helpers. Then there's staff meetings; the people who are using text are always at a disadvantage, because speaking is so much faster at getting your point across than text, and people naturally miss what you say if they aren't looking. For example, I was almost always writing notes for meetings when I was staff (meaning, literally on a separate page and listening to the meeting while typing) and yet I was still one of the only people calling out the things that people said in text. It's just impractical. Then there's the rare, but still present situations where you need to handle an emergency with a player or other staff and the best way to do that is through speech.
So TL; DR: to answer your question, no. You can't be (or apply for) staff without a working mic and a Discord.