The relativistic chiral EFT approach is introduced to improve the existing estimates of the $nn\rightarrow ppee$ amplitude by including contributions up to next-to-leading order. The estimated $nn\rightarrow ppee$ amplitude, $|\mathcal{A}_{0\nu}|=0.0209(7)\ {\rm MeV}^{-2}$ at the kinematics $|\boldsymbol p_i|=25$ MeV, $|\boldsymbol p_f|=30$ MeV, can be used to determine the leading order decay operator in nuclear-structure calculations.The approach is also applied to calculate the ground-state-to-ground-state $nn\rightarrow ppee$ matrix element at $m_\pi\simeq 806$ MeV on a finite-size lattice, showing promising agreement with the recent first lattice QCD simulation.