if youre in the usa, it should (?) be a legal requirement that your job provide "reasonable accommodation" including things like assistive equipment (seating is common), changes in job duties, etc. as much as the process can suck, i can't see any reason not to try it, especially if you're active with medical providers and wanting to keep the job... if you live in an at-will state, it's possible they can try and fire you for it by coming up with some other reason, and if they did... a. is that really an employer you would have been able to continue working for? and b. you can take legal action, with compensations up to and including getting your job back, if you can prove it was related to your disability. i don't think any of this should be relevant, especially with a major employer, but if it became relevant you're not just SOL.
outside of the usa, i'm not so sure, but i can only imagine that most places would have some sort of system in place (and likely one better than here). at the very least, it should be alright to start the conversation regardless of where you are, and your employer can give more specific advice on what that process may look like. you could even do something like try doing the job with a wheelchair - something that would allow you to sit to help with the dizziness and prevent a lot of potential injury from dislocation.
if it comes to the point that you decide to leave your job and seek ssdi disability (usa-specific) rather than employment doing something less physically taxing, i have some words of caution. first of all, find a lawyer BEFORE you start your claim. most ssdi lawyers know that disabled people don't have much money to throw around, so they work on contingency. it will help your claim go faster and easier, and they can help you navigate the jungle that is applying for ssdi. second, make sure you have as much info as you can before applying. what you did to try and make the job work, resume-like info about your past jobs, names, phone numbers, addresses of all the doctors youve seen since the onset of your disability, all your diagnoses, etc. the more you have everything in order, the less of a nightmare it'll be compiling the ridiculous amount of information they ask. getting a copy of your medical records doesn't hurt your chances either. and third... make sure you have a way to provide for your living expenses, without working for income, for the next 2-3 years. help from family, even moving back in with them, is the best option. savings can help but i have a hard time imagining anyone our age would have enough in savings to live for more than a couple months, much less a couple years... i'm at 1 year 9 months since my initial application and don't have a hearing date yet.
best of luck to you