If celebrities visit public places, such as malls and bars, people
will crowd them demanding for a picture or autograph.
1: Why did the writer use the "-ing" form of the verb "demand"? Is there any rule for this?
2: Why there was no article before the word "autograph"?
Demanding for a picture or autograph is a participial construction used as an adverbial clause (containing an adverbial participle) modifying will crowd to express the reason why people will crowd the celebrities, though others might argue that this is an adverbial of purpose.
A picture or autograph: the indefinite article an was omitted before autograph probably to include picture and autograph into one category: the type of thing fans may ask for.