I actually need something weaker than this but my hope is that this holds in its fullest generality. Let $I$ be a small diagram and $I'$ a full subcategory of $I$. Let $F: I\to {\rm vec}$ be a functor to finite dimensional vector spaces and $F': I'\to {\rm vec}$ the restriction of $F$ to $I'$. Assume that $F \cong \oplus_i F_i$ and thus $F'\cong \oplus F'_i$ where $F'_i$ is the restriction of $F_i$ to $I'$.
Since $\varinjlim F$ is a cocone for $F'$ we get a unique morphism $\varinjlim F'\to \varinjlim F$ by universality.
Clearly I have canonical isomorphisms $\varinjlim F \cong \oplus_i \varinjlim F_i$ and similarly for $\varinjlim F' \cong \oplus_i \varinjlim F'_i$. But is this natural with respect to restrictions?
I.e., do I have a commutative diagram
$$\require{AMScd} \begin{CD} \varinjlim F' @>{\cong}>> \oplus_i \varinjlim F'_i\\ @VVV @VVV \\ \varinjlim F @>{\cong}>> \oplus_i \varinjlim F_i; \end{CD}$$
PS: you may assume that the there are finitely many $F_i$.
Normally this kind of thing is verified by the "what else?" argument, that is, what are the chances you really have two different canonical maps in that square? That said, it could be comforting to know it's possible to really check such a thing. So:
Let me change your $\oplus_i$ to $\oplus_j$, to avoid collision of notation. We have to show that for every $i'\in I'$ and every $j$ in the direct sum, the two maps $F'(i')\to \varinjlim F'_j\to \varinjlim F_j, F'(i')\to \varinjlim F\to \varinjlim F_j$ coincide. (To reduce to $i'$, I use the universal property of the colimit, and to $j$, the fact that a coproduct of finite dimensional vector spaces is necessarily finite, hence a product. It would be easy to handle infinite coproducts in an abelian category using that they're still subobjects of the product.) The first map is $F'(i')\cong \oplus_j F'_j(i')\to F'_j(i')\to\varinjlim_{I'} F'_j\to \varinjlim_I F_j$ and the second, $F'(i')=F(i')\to \varinjlim F\cong \oplus_j \varinjlim F_j\to \varinjlim F_j$.
See the image below for the appropriate diagram. The upper left squares and the upper right triangle commute automatically because they have branches that are equalities-just use the same map on each side. The lower left square and the lower right triangle commute by the theorem that a colimit is unique up to unique isomorphism preserving the universal cocone, since the $\gamma$ maps here are legs of the cocone. Finally, the central bottom square commutes by the definition of $\oplus_j \gamma_{j,i'}$.