Step 1: Recall the definition of homoleptic and heteroleptic complexes.
A homoleptic complex contains only ONE type of ligand coordinated to the central metal atom. A heteroleptic complex contains TWO or more different types of ligands.
Step 2: Analyse option 1 - $[Co(NH_3)_4Br_2]^+$.
This complex has two types of ligands: $NH_3$ (ammonia) and $Br^-$ (bromide). Since there are two different ligands, this is a heteroleptic complex.
Step 3: Analyse option 2 - $[Co(C_2O_4)(NH_3)_4]^+$.
This complex contains $C_2O_4^{2-}$ (oxalate) and $NH_3$ (ammonia) as ligands. Two different ligands, so this is heteroleptic.
Step 4: Analyse option 3 - $[Co(NH_3)_6]^{3+}$.
This complex has SIX ammonia ($NH_3$) ligands and NO other ligand type. Since all ligands are the same (only $NH_3$), this is a homoleptic complex.
Step 5: Analyse option 4 - $[Co(CN)_4Cl_2]^-$.
This complex has $CN^-$ (cyanide) and $Cl^-$ (chloride) as ligands. Two different ligands, so this is heteroleptic.
Step 6: Identify and confirm the homoleptic complex.
Only $[Co(NH_3)_6]^{3+}$ qualifies as homoleptic since it contains exclusively $NH_3$ as the ligand. This is option 3.
\[ \boxed{[Co(NH_3)_6]^{3+} \text{ is homoleptic}} \]