Step 1: Determine the atomic number of molybdenum (Mo). Molybdenum has an atomic number of 42, meaning a neutral Mo atom has 42 electrons.
Step 2: Write the electron configuration for the neutral Mo atom. The ground state electron configuration is $1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6 5s^1 4d^5$.
Step 3: Consider the ion Mo$^{3+}$. This means the atom has lost 3 electrons. Electrons are removed first from the outermost shell, which is the 5s orbital, then from the 4d orbital.
Step 4: Remove electrons accordingly: first remove the single electron in the 5s orbital, then remove two electrons from the 4d orbital. This results in $5s^0 4d^3$ for the valence electrons.
Step 5: Combine the core electrons with the valence electrons to write the full electron configuration for Mo$^{3+}$: $1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6 5s^0 4d^3$.