Think I am depressed and need to quit, need advice on managing exit appropriately

I would recommend instead of quitting, use your current health insurance to seek help of a therapist via telehealth. While I agree with everyone your mental health is your primary concern and it should be, realistically speaking other things like lack of funds, lack of insurance will soon come into play and would be in a downward spiral.

Remember, can’t escape but can learn how we react to situations and sometimes it’s just a matter of retraining your brain.

Agreed - your health is the most important. But try to find another job before you leave ideally. If that is too hard, communicate with your staffer that you are going through a rough time and need some slack and then try to interview for other places with the decreased workload. If that doesn't work, these banks have programs in place where you can request time off to seek mental health help. Maybe during that leave of absence you could set up some calls / network with other places. You could also have your doctor write a note about you having anxiety issues/symptoms etc. They will not fuck with you if you have a medical note as they can be sued tremendously as it could count as discrimination based on health.  

Honestly, if I were you I’d talk to a doctor about this all and see what the doctor thinks about taking an extended leave. I’d imagine the doctor would 100% recommend you take this time off.

Then, tell your company that you've been facing some health issues lately and you can tell them that your doctor has recommended you take some time off to focus on your health.

Just keep things vague & be assertive. Don't "ask" for time off, rather tell them you need to follow your doctor's orders and cannot fulfill your work responsibilities until youve had some time to sort things out.

I realize this plan isn't perfect, but i think it's a quick way to clear your plate and step away from work without attracting too much attention. You can then use your time off to heal & find a new job.

Best of luck

Was in a similar boat. A few big questions:

• did you apply round 1? If yes, you really are only holding out until interviews in the next few months unless you really believe your recommenders would be so vengeful they would try to go around you (seems pretty messed up, but people can be that way)
• Have you used all your vacation? Take a week off and you might feel better. If you haven’t used all your vacation, do that first before making a decision.
• Ask yourself if your current work situation is worse than 2 months ago (are you working with assholes currently or on a bad project?) 
• If you know you don’t want to be in the industry, it’s possible to use it as a rational pivot in an industry. I debated doing this. Here’s a possible explanation, “I 100% knew the industry wasn’t for me/ I didn’t want to come back, and wanted an alternate avenue to pursue prior to bschool. I’ve only known 2 jobs and figured something in the interim between now and September could give me an alternate perspective that would be very helpful for furthering my understanding of where I want a career. I also knew that there was no chance I would be able to search or interview effectively while in my current role” 

Another strategy if you are just trying o buy time is to go on leave as other people mentioned. It’s scary, but you need to do what you need to do and in the long run if you know this isn’t the job or even field you want f them. Go to your manager and say, “I’m having serious mental health issues, I’m taking 2 weeks off.” They might push back and say something like, “this is what you signed up for, you need to do this etc.” your reply, “I understand this isn’t ideal. I don’t want to be in this situation either, unfortunately I am past a point of recovery. I am taking two weeks off and I would like to return to the role after that time” Potentially they will say, “if you leave you are fired” in which case, you work for assholes and have bad managers and gtfo.

Edit: just adding to the above, a rational thing for a manager to do is to give you the two weeks if you have been there longer than a year and are at least an ok performer. Training a new hire blows and you being gone 2 weeks is way better than losing you completely if you are even decent at your job.

How much money do you make? 

School considerations: I've never heard of this happening and think you should not be worried about it. I can't imagine somebody being so vengeful that they'd go to an MBA program and ask for them to rescind a student's admission because he/she quit a few months earlier than the boss wanted. And even then, I don't see the MBA program following through on it (unless maybe there was an eight-figure donation on the line). You'd need to have lied on your application, committed a felony, or something similar. 

Future Job Considerations: I haven't needed a recommendation letter for any job I've ever applied for and again, even if you need one, I don't see quitting a few months earlier than you otherwise would as a reason somebody would give you a bad recommendation. You know your firm's leadership better, so you be the judge of how vengeful they are.

Closing: Plenty of people take off a lot of time before starting at business school. Your applications are in. If you want to stick it out through interviews so you feel like you're being 100% truthful about your employment status, go for it, but I feel like you can quit soon if you want to and still feel good about your applications. HR isn't going to call up an admissions officer and say "oh hey this kid quit". As long as you feel confident that you're going to get into a program you want to actually attend, go quit. Take care of yourself. Unwind for a bit. Get whatever help you may feel like you need. You don't have to tell anybody at your firm why you're really leaving if you don't want to... just say that you're quitting to go to business school and you're going to travel for a while beforehand or something like that.

What really strikes me in your post is the degree to which you have ceded control over your life to your job which seems beyond typical/unhealthy and is something you have to address not just today but in the future. Of course the finance industry takes over your personal life and all that but you're literally afraid of exercising your agency to leave your job (rather than the prospect of being without said job). That is an extreme situation to be in and not normal. You entered into an employment contract not indentured servitude.

Is your existing team abusive and retaliatory? If not, then what are you afraid of? If yes, then even more reason for you to find a way to leave your situation sooner than later.

There are some helpful tips here on how to time your departure with your MBA admission process (if you think you can wait a few more months) but in any case, I highly recommend you to quit as soon as you have you acceptance. Not just for your present mental health but also to develop the essential life-skills of pushing back and saying no at work (and being able to to quit at any time is a critical part of establishing those boundaries).