A spin structure on a principal $SO(n)$-bundle $E$ is a cohomology class in $H^1(E, \mathbb{Z}/2)$ that restricts to a generator of $H^1(SO(n), \mathbb{Z}/2)$. It is well-known that an oriented vector bundle admits a spin structure iff $w_2 = 0$.
Let $V$ be an oriented vector bundle. Then it is easy to see that $V \oplus V$ is also oriented, and $w_2(V \oplus V) = 0$. Thus $V \oplus V$ admits a spin structure.
Is there a canonical spin structure on $V \oplus V$? If so, what is it?
Yes there is one.
The transition functions for $V \oplus V$ lie inside the diagonal of $SO(n) \times SO(n) \subset SO(2n)$. Because the map from each factor $SO(n) \to SO(2n)$ induces the identity on $\pi_1$, that means that the composite of $g: SO(n) \to SO(n) \times SO(n) \to SO(2n)$ (first map the diagonal) induces zero on $\pi_1$. There are thus two lifts of $g$ to a map to $Spin(2n)$. Only one of them is a homomorphism: the one with $\widetilde g(1) = 1$.
Now given a bundle $V$, apply $\widetilde g$ to the transition functions to get a spin structure on $V \oplus V$.