so one uses equations of motion to describe liquids (e.g. Navier–Stokes equations). These are equations for $\vec{v}(\vec{r},t)$ with boundary conditions on the surface $S$ of the liquid (e.g. $\vec{v}(\vec{r}\in S,t) = \vec{0}$).
How should one incorporate surface tension $\sigma$ in these equations/boundary conditions? It seems, only boundary conditions must change, and $\Delta p = \sigma (1/R_1 + 1/R_2)$ is the first thing that comes to mind, but how to get $1/R$ from $\vec{v}(\vec{r},t)$?
You don't want $1/R$ (although technically it means the same) but rather the full curvature term: $\Delta p=\sigma \kappa$. In fact you will get a source term in the Navier-Stokes equations that looks like this: $$\sigma \kappa \delta(n) \mathbf{n} $$ where $\delta(n)$ is the Dirac Delta function that only has a value at the interface and $\mathbf{n}$ is the interface normal. The curvature $\kappa$ can be written as the divergence of the unit interface normal: $$\kappa=\nabla \cdot \mathbf{\frac{n}{|n|}} $$
Apart from the source term you indeed also have boundary conditions on the interface which are basically the standard free slip condition and a jump for the normal stress coming again from the Laplace pressure. There is a good explanation of these in the first part of the seminal work on fluid-fluid CFD by Brackbill.
If you are interested in the curvature itself, I think Slides 22-28 of this course on wetting are probably also a good source to take a look at for more background.
