When is a vector field on a manifold restricted to a submanifold $X$ a vector field on $X$? Let $X$ be an embedded submanifold of $M$ and let $V$ be a vector field on $M$. One can restrict $V$ to $X$, but it may not define a vector field on $X$. 
Example: The vector field $x^i\partial_i$ on $\mathbb{R}^n$ does not define a vector field when restricted to $S^{n-1}$. 

Given a vector field $V$ on $M$ and an embedded submanifold $X$, is there a way to determine whether $V|_X$ is a vector field on $X$?

One can define the projection of the restricted vector field. Choose a Riemannian metric on $M$ and let $NX$ be the normal bundle of $X$ in $M$, then $TM|_X = TX\oplus NX$. Let $p : TM|_X \to TX$ be projection onto the first factor, then we obtain a vector field on $X$ given by $p\circ V|_X$. Note, this does not depend on the choice of Riemannian metric.
If we were to perform this construction for the above example, the vector field we obtain on $S^{n-1}$ is the zero vector field. It is clear that $V|_X$ is a vector field on $X$ if and only if $p\circ V|_X = V|_X$. This doesn't satisfy me though as it is rather tautological.
 A: First and most importantly forget about Riemannian metrics: they are an extraneous structure which will only lead to confusion, weeping and gnashing of teeth.      
That said, suppose for simplicity that $X\subset M$ is given as the zero set of a smooth function $f:M\to \mathbb R$ satisfying $df(x)\neq 0$ for all $x\in M$.
Given a vector field $V$ on $M$ its restriction to $X$ will be tangent to $X$ if and only if for all $x\in X$:   $$\langle df(x), V(x)\rangle =0              \quad (\bigstar)$$ 
In local coordinates $y_1\cdots ,y_n$ for $M$ the condition becomes  $$\langle \sum_i\frac {\partial f(x)}{\partial y_i}dy_i,  \sum_j v_j(x)\frac {\partial}{\partial y_j} \rangle= \sum_i\frac {\partial f(x)}{\partial y_i}\cdot v_i(x) =0 \quad (\bigstar \bigstar)            $$ Let's check on your example, where $M=\mathbb R^n, f=(\sum_i y_i ^2)-1 $ and $V(y)= \sum y_j \frac {\partial}{\partial y_j}$.
The condition $(\bigstar \bigstar)$ becomes $\sum_i 2x_i\cdot x_i=0$, which is clearly false at every point $x$ of the sphere (since there $\sum_i 2x_i\cdot x_i=2$), as it should since the global tangent field $V$ on $\mathbb R^n$ certainly does not restrict to a tangent field of the sphere.  
In general the submanifold $X$ will be given by several equations $f_k=0$ where the   $f_k$'s are local equations for $X$, but the generalization of $(\bigstar \bigstar)$ to the more general situation is straightforward.
