Monolayer WTe2, a centrosymmetric transition metal dichacogenide, has recently been established as a quantum spin Hall insulator and found superconducting upon gating. Here we study the pairing symmetry and topological nature of superconducting WTe2 with a microscopic model at mean-field level. Surprisingly, we find that the spin-triplet phases in our phase diagram all host Majorana modes localized on two opposite corners. Even when the conventional pairing is favored, we find that an intermediate inplane magnetic field exceeding the Pauli limit stabilizes an unconventional equal-spin pairing aligning with the field, which also hosts Majorana corner modes. Motivated by our findings, we obtain a recipe for two-dimensional superconductors featuring "higher-order topology" from the boundary perspective. Generally, a superconducting inversion-symmetric quantum spin Hall material whose normal-state Fermi surface is away from high-symmetry points, such as gated monolayer WTe2, hosts Majorana corner modes if the superconductivity is parity-odd. We further point out that this higher-order phase is an inversion-protected topological crystalline superconductor and study the bulk-boundary correspondence. Finally, we discuss possible experiments for probing the Majorana corner modes. Our findings suggest superconducting monolayer WTe, is a playground for higher-order topological superconductivity and possibly the first material realization for inversion-protected Majorana corner modes without utilizing proximity effect.