How do you create a constructor that use a vector as an argument and parameter?

Two problems:

1. Don't pass function arguments by value, pass it by const reference instead. The only exceptions are (a) when it is a built-in type (int, char, double), or (b), if the parameter is meant for output (in which case it should be passed by reference, without const)

2. You've defined the variable to receive the number definition locally within the function, rather than as a class member variable.

Also, you can take advantage of the copy constructor of class vector and just initialize the entire vector in the initializer list without bothering with a loop:

class BigInteger {
   std::vector<int> number_;
public:
   BigInteger(const std::vector<int>& num) : number_(num) {}
};

P.S.: as Sergey pointed out in the comments, having such a constructor may be pointless: whenever you want to initialize a BigInteger, what you have is either a (decimal) text representation of a long number, or just a simple integer type. So you should provide constructors for those types instead.

The only reason to provide a constructor for a different type would be if you want to interface with another kind of big integer class - in which case the question arises why you need a second one.

In C++11, the constructor that you've written can be written like so -

BigInteger(const std::vector<int>& num)
{
    decltype(num) number{num};
}