I found a thing called Chemlambda. It is a graph-rewrite system capable of evaluating (most) lambda calculus expressions using only rewrites involving up to four nodes, using a technique resembling DNA replication.
Here is the page: http://chorasimilarity.github.io/chemlambda-gui/
Here is a (work in progress) Javascript sandbox I made for it: https://jsfiddle.net/ishanpm/0k98ceha/show
There are some glitches with it in its current form, for example when replicating lambda abstractions that ignore their input, but it's still a very interesting approach to computation.
Side note: Are other types of automata, such as this graph-rewrite automata, considered on topic in the main board?