So, you wanted etymology, right? Well, every words you wrote is sino-Korean, which means it is composed with Chinese character.
Let's start with 사원. 사원 is 社員 in Chinese character. 社 means 'to meet', and it is also 사 of 회사. 員 means 'number of people'(인원), and it can be used as meaning of member. (조직원 : gang member)
Next is 주임. At Korean dictionary, it says it means '직장, 단체 따위에서 어떤 일을 주로 담당함. 또는 그런 사람'(rough translation : mostly in charge of some work at job or organization. Or one who do that.). It is 主任 in Chinese character. 主 means 'owner', but it is also used at word 주로, means mostly. 任 means 'assign (some work)', and it also used in 임무, means assignment or duty.
대리 comes next. It is exactly same word with 'substitution', 'deputy', or 'proxy'. 代理 in Chinese character. 代 means 'replace;, and 理 means 'govern' or 'rule', and maybe 'process some work'. At dictionary, it says '은행이나 회사 따위의 집단에서 부장, 지점장, 과장 등의 직무를 대신하는 직위. 또는 그 직위에 있는 사람. 흔히 유사시 과장의 직무를 대신할 수 있는, 과장 바로 아래의 고정된 직위를 이르기도 한다.', roughly translate, it means 'position that can substitute 부장, 점장, 과장's duty at the bank, company, etc. Or one who at that position. Normally, it also means position directly lower than 과장, who can substitute duty of 과장 at emergency'.
Next is 과장. Before we talk about 과장, we have to know what is 과. It is one of department, normally lower than '부' or '실'. 課長 in Chinese character. 課 is 과. (Just for information, it means to study or process.) And 長 normally means 'long', but it also means chief of something. So 과장 means chief of 과.
Same thing with 부장(chief of 부部 : 'group' or 'command'), 실장(chief of 실室: house. Also means chief of organization end with 실 like 연구실.).
차장 is somewhat different. 차 is 次, means 'to be second to'. So it means 'Position that is second to 장. Or one who at that position.'
수석 means 'First place'. 首席, and 首 means 'head', 席 means 'seat'. So 수석부장 is 'highest 부장'.
이사 means 'Executive organization which processes office work of corporate body and takes legal action. Or one who at that position'. 理事 in Chinese character. I talked about 理 at 대리, and 事 means 'work'. I don't know how this became that.
상무 is same with 상무 이사 (or 상무위원 if it is public organization, and I'll skip it.). 상무 itself means 'Everyday work', and 상무 이사 means 'Organization which processes normal works at some organization or company. Or one who does it.' 常務, 常 means 'always', and 務 means 'labor for something'.
전무 means 1. Expertly handles some job. Or one who does it', and 2. '=전무 이사'.(One of 이사 of company. Direct work of company while assist 사장 and 부사장.). 專 or 專務 means 'only', and 務 is same with 상무.
사장 is chief of 사社, which is 회사. Same 사 with 사원.
Then, what is 회장? So I don't think it is just simply 'chief of 회', although Chinese character version means it. 會 is 회 of 회장, means 'to meet', and also 회 of 회자. In Korean dictionary, it says it is 'position higher than 사장. Normally it means chief of corporation, etc.'
Sorry for bad English, and there might be wrong translation/wrong description. But if you have any question, please reply.