Will they take away my offer?

So I have a conditional offer to study medicine for 2024 September but I’ve recently got diagnosed with depression and struggling a bit. I searched it up the uni website and they said “A request to defer once an offer of a place has been given for the current admissions cycle cannot normally be considered.” So should I still try to email and ask but will they take away my offer for 2024 September?

While they might not revoke your offer for contacting them, it's always a possibility. However, in most cases, universities appreciate open communication, especially regarding health concerns

So I have a conditional offer to study medicine for 2024 September but I’ve recently got diagnosed with depression and struggling a bit. I searched it up the uni website and they said “A request to defer once an offer of a place has been given for the current admissions cycle cannot normally be considered.” So should I still try to email and ask but will they take away my offer for 2024 September?

They won't revoke the existing offer for asking. Even if a deferral is not possible, you'll still have the offer for 2024 if you want it.
Note that they say that a deferral is not ordinarily possible, so I would provide evidence of your diagnosis in support of your request.

Some Unis can be reluctant about Medicine deferrals for a traditional gap-year / travel etc , but health concerns are entirely different and Unis will try to be helpful in these circumstances. They certainly wont 'cancel' your offer.
Call the Uni (not email, it takes to long to get a response) and explain.
They may want a doctor's letter, or a letter from your school, so do this as soon as you can.