I have attempted to prove this but am unable to complete the proof. Below is my attempt.
Let $\mathcal{A}$ be a category satisfying the conditions in the title and $\{M_i,\phi^i_j\}$ be a finite direct system. For each $i\leq j\in I$, let $M_{ij}=M_j$. Then we have a map $\mu_{ij}:\bigoplus M_i \to M_{ij}$ induced by the maps $\phi^i_j:M_i\to M_{ij}$, $-1 : M_j \to M_j=M_{ij}$ and $0:M_k \to M_{ij}$ for $k\neq i,j$. Combining all these maps we get a map $\mu=\prod\mu_{ij}: \bigoplus M_i \to \prod M_{ij}=\bigoplus M_{ij}$ since the index set is finite. Let $(S,i)$ be the kernel of $\mu$ where $i:S \to \bigoplus M_i$. Let $p:\bigoplus M_i \to D$ be the cokernel of the morphism $i$. Then I expect $D$ to be the required direct limit. However I am unable to prove the universal property. Is it possible to proceed along these lines and complete the proof ? If not how does one prove that finite direct limits exist ?
Note that the result is true in any category (not necessarily additive) having finite (co)products, (co)equalisers for any pair of parallel of arrows and a terminal (initial) object. More generally again, a category admitting arbitrary small (co)products and (co)equalisers for any pair of parallel arrows is (co)complete. This is a completely standard fact and you can find a proof for it, for example, looking at Theorem 2.8.1 and Proposition 2.8.2 of Handbook of Categorical Algebra, Volume I by F. Borceux.