When you need a new office chair at work the best way to ask for a new chair for work is to arrange a meeting with your boss or manager.
Explain to the boss or manager in the meeting that you need a new office chair and if there's a good reason to replace your office chair you will usually be able to get a new office chair.
If the boss or manager will not buy a new office chair for you then consider asking your boss or manager if it's okay to bring your own office chair in.
Sometimes you will be allowed to buy your own office chair and bring it to the office.
I did this before when I was working in an office and the chairs we had were cheapo office chairs with no arm rests and very poor seat padding.
I bought a good office chair from Ebay and brought it in but I had to put my name on it by engraving it into the chairs cylinder.
Then I could take the office chair home with me when I got a new job.