OK, you need to recognize two similar-sounding but quite different uses of "account". The first is typified by a sentence like:
"Alexander Selkirk met Daniel Defoe, and gave him an account of his life as a castaway sailor."
Here, Alexander is simply relating what happened. This is the most usual use of "account" in everyday speech unless you happen to work in company finance, and it is the construction which will come first to mind for most people.
Contrast that with a different sentence:
"Mr Robbins was called before the enquiry, and asked to account for his actions on the day of the accident".
The difference here is that Robbins is not asked for a mere narrative: he has responsibility for his actions and is being asked to show that he discharged that responsibility properly. It is in that context that "for" is more appropriate than "of".
A cashier has been placed in a position of trust and responsibility, and therefore is asked to account for the money and other valuables in their care. The particular sentence you have been asked to complete, tells us a little more, though. "A cashier is liable to render account..." - this is highly formal language which would almost never be used in contemporary British English.
It could be interpreted either as the language of a legal or quasi-legal document, or the formal language of business, of perhaps a half-century ago. So, today in common use most people would, I suspect, use "of", even in relatively formal circumstances, and even if they carry responsibility for the matter.
"We had our club meeting last night, and Sarah our treasurer gave an account of the finances. They are not too healthy."
But there definitely comes a point where "for" will be used to reflect and emphasize responsibility.
"We had our club meeting last night, and Sarah our treasurer gave an account of the finances. A question from one of the members asked her to account for the $500 deficit in our funds."
In exactly the same way, the sentence you are asked to complete has a meaning carried by that particular wording. It is not just saying that a cashier must be able to present accurate records of money paid in and out -- it is saying that the cashier has responsibility for the money in their care and must be prepared to account for its proper handling. In this context, "for" is definitely the correct word.