The most common and most likely answer to this crossword clue is the 7 plus 5 letter word RUNNING BOARD
I've seen this crossword clue in The Times
R
U
N
N
I
N
G
B
O
A
R
D
Buddy explains!
The straight part
We know that "Part of early motor car/the chairman’s responsibility" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case part of early motor car/the chairman’s responsibility = running board.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.