How would I go about requesting a hardship discharge?

I am so sorry for your loss. I would start with a meeting with your chaplain. I am sure if that is what you want/need they can walk you through the process.

With the amount of time it can take to process voluntary separations you are probably better off just waiting it out. It is possible to get a voluntary separation submitted, signed and approved in a month or two, but that's when you dig into the reg and walk it through yourself. Even then, you're usually going to get it approved for 90 days later. So, you walk a separation packet through in a month, you get out 90 days later and you've only saved yourself two months after all that work and headache

Sorry to hear of your loss. If I were in your shoes I would go see the Chaplain. I saw them do things I didn't think possible when I was in the Corps. No matter what you know the governments old saying, "Hurry up and wait!"