If your working definition of "vector" is "an ordered list of numbers", then you can start there.
Let's consider $v=(A,B,C)$ where $A,B,C$ are all real numbers. This gadget would be a vector for us. How could one look upon it as a function? Well we could name the positions #1, #2, and #3, and say $v(1)=A, v(2)=B, v(3)=C$. Therefore the domain (things for $i$ in $v(i)$) is $1,2,3$ and the output is determined by whatever is in that position.
Now let's think of a sequence of real numbers $a_1, a_2,\dots$. If I dress them up: $v=(a_1,a_2,\dots)$ this is also a (very long) vector. To think of it as a sequence we can pull the same trick of using the $i$'th position to record the output for $i$, and so $v(i)=a_i$ has given you a function from $\mathbb{N}$ into $\mathbb{R}$. Really, $a_i$ was already in functional notation, and you probably wouldn't need to introduce $v(i)$, but I'm making a point here :)
But why should we stop with a finite set or $\mathbb{N}$? Could we use $\mathbb{R}$ as subscripts for a vector? Sure, except now that the inputs for $i$ are not discrete anymore, it becomes impossible for us to physically write a vector with continuum many positions. But we can still imagine it. Just imagine you have a bunch of positions, each labeled with a real number $r$, and the whole bunch are bracketed by parentheses (). In the $r$'th spot, you put the output of $v$, and so we would write $v(r)=Q$, where $Q$ is that particular output.
Really, order isn't even necessary. Given any function from a set $X$ into a set $Y$ you just imagine a bunch of positions, one for each thing in $X$, surrounded by parentheses. In each position you are allowed to fill in a thing from $Y$. Let's call the whole vector $f$. What is $f$'s value at $x\in X$? Well, whatever value of $y$ is in the $x$th position of course! We denote this with $f(x)=y$.
In this way $f$ can be thought of as a very long vector with entries indexed by $X$, and each entry contains the value of $Y$ which is the output.