The answer is **no** under the additional assumption that the anticanonical system of the $3$-fold $V$ is ample, i. e. that $V$ is Fano (in a precedent version of the answer I wrongly assumed that this was implicit in the question, see the comments below).

By the sake of simplicity let us assume that $-K_V$ is very ample, for the general case you can look at [this][1] paper by A. Beauville.
 
If $V$ is a Fano 3-fold containing $S$ as a hyperplane section, by the  Lefschetz Hyperplane Theorem the restriction map $$r \colon \textrm{Pic}\,V \longrightarrow \textrm{Pic} \, S$$ is injective, and this is a constraint on the $K3$ surface $S$.

For instance, it is well-known that for all $g \geq 2$ there exists a polarized K3 surface $(S, \, L)$ of genus $g$ such that $\textrm{Pic} \, S$ is generated by $[L]$, see for instance 

http://mathoverflow.net/questions/124880/picard-group-of-a-k3-surface-generated-by-a-curve

On the other hand, we know by the work of Iskovskih that smooth Fano threefolds with $b_2=1$ form a bounded family (they belong to exactly $17$ isomorphism classes). 

Therefore, if $g$ is large enough, every smooth Fano $3$-fold $V$ of genus $g$ satisfies $b_2(V) \geq 2$, i.e. $\textrm{Pic}\, V $ has rank $\geq 2$. Since $\textrm{Pic} \, S$ has rank $1$, by the injectivity of $r$ it follows that $S$ *cannot*  be obtained as a hyperplane section of a smooth Fano $3$-fold.

Let me finish by asking the following

**Question.** Is there any example of a smooth $K3$ surface $S$ appearing as a *non-ample* anticanonical divisor in a smooth 3-fold $V$?

  [1]: http://math1.unice.fr/~beauvill/pubs/Fano.pdf