Can you entangle two already existing particles?

All scattering experiments of high energy physics start with the scattering of two particles. At the point of interacion there exists a quantum mechanical wavefunction describing mathematically the interaction, and all particles coming out of the interaction are by definition "entangled" by conservations laws and quantum number conservations. See this answer of mine..

Any interaction between two systems will typically lead to entanglement. If $vert phi_1rangle$ and $vert psi_2rangle$ are eigenstates of $H_1$ and $H_2$ respectively, then an interaction term $V_{12}$ will usually be so that the eigenstates of begin{align} H=H_1+H_2+V_{12} end{align} are not of the form $vertchi_{12}rangle =vertphi_1ranglevertpsi_2rangle$ but rather of the form begin{align} vertchi_{12}rangle =sum_{ij}c_{ij}vertphi_iranglevertpsi_irangle, . end{align} Loosely speaking, suppose begin{align} Hvertphi_1ranglevertpsi_2rangle &=(H_1+H_2+V_{12})vertphi_1ranglevertpsi_2rangle, , &=(E_1+E_2)vertphi_1ranglevertpsi_2rangle +sum_{ij}vertphi_irangle vertpsi_jranglelangle phi_i;psi_jvert V_{12}vertphi_1;psi_2rangle, . end{align} Unless $langle phi_i;psi_jvert V_{12}vertphi_1;psi_2rangle$ is proportional to $left(sum_{i}c_ivertphi_irangleright)vertpsi_2rangle$, the result will not be a separable state. But if $V_{12}$ gives such a linear combination, there is no interaction with the 2nd system as it has remained undisturbed by $V_{12}$.

This is not a rigorous argument but it does give insight into how entanglement typically results from interactions.