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We can surely have such a function. Suppose we are in working in $\mathbb{R}^2$. And, define the function: $$\vec{\mathbf{v}}(t,u)=\langle t+u, t^2+2u+1\rangle$$ The question arose in my mind when I tried to graph this. I couldn't. Suppose I had only one variable, say $t$, the direction of motion of the parametric curve would be along increasing $t$. However, in this case, how can we decide what the direction of motion will be? Or, in simple words, how do we graph this function?

I have another doubt. In $\mathbb{R^3}$, vector functions of one variable represent curves, and vector functions of two variables are surfaces. Can we have this distinction in $\mathbb{R^2}$ too? Or more generally, in $\mathbb{R^n}$, can we plot a vector function of $n$ or $n+1$ variables? And what distinction does it bring about in the nature of the plot?

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    You can only represent that function (in the usual way) in $\mathbb{R}^4$. Notice that the graph of a function consists of points $(x,f(x))$. – 2017-02-14
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    Regarding the second part, you don't use vector function of two variables to represent surfaces in $\mathbb{R}^3$. – 2017-02-14
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    @Smurf Re: your second comment, perhaps OP (implicitly) means the way we regard a curve or surface as the _image_ of a parametrization, so that a surface is represented by (the image of) an $\mathbf{R}^{3}$-valued mapping. – 2017-02-14

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The graph of a function $f: X\to Y$ is defined as the set $$\{(x,y)| f(x)=y\}$$

Notice that $V$ is a subset of the set $X\times Y$, which is why the graph of a single variable function $f:\mathbb R\to\mathbb R$ is a subset of the plane $\mathbb R^2$.

Following this, the graph of your function would be a subset of $\mathbb R^4$, and sadly, the nature of our reality prevents us from being able to draw a $4$-dimensional structure.

As for your other doubt, first a warning:

vector functions of one variable can represent curves, but they are not equal to curves. Same with surfaces. There can be two different vector functions that represent the same curve/surface.

But that said, yes, this can be generalized so that functions from $\mathbb R^k$ can be represented by subsets of $\mathbb R^n$ for $n>k$.