I googled "lattice of convex sets"+distributive and the second link led me to the abstract for the paper: "Geometric Condition for Local Finiteness of a Lattice of Convex Sets" in Mathematica Moravica, Vol. 1 (1997), 35–40 by Matt Insall, which contains the following sentence:
We give elementary examples which establish the following facts: the lattice of convex subsets of a Hilbert space is not locally finite, it is not modular (hence not distributive), and locally finite lattices of closed convex sets in any Hilbert space have very restrictive geometric arrangements of their members.
This means that the answer to your question is no, since $\mathbb{R}^2$ is certainly a Hilbert space.
Unfortunately, the journal did not have the paper online, but I was able to find Insall's webpage and from there, the PDF for the paper.
The paper is relatively easy to read and has illustrations for the examples.