Given $N$ vertices, I am interested in the distribution of the size of connected components of the random graph formed by assigning $M$ edges to randomly chosen pairs of vertices so as to form a simple graph (i.e. no self-edges / multi-edges permitted).
Most literature seems to focus on the classical Erdős–Rényi random graph, so I had no luck even finding the name of a random graph model like this.
The name for this model is the Erdős–Rényi random graph and it is denoted usually as $\mathcal G(n,m)$. This is actually a probability space where each outcome (a graph $G(n,m)$ with $m$ edges on $n$ labeled vertices) is assigned the probability $$ {\rm P}\{G(n,m)\}=\frac{1}{\binom{\binom{n}{2}}{m}}\,. $$ This model has very close properties to the model $\mathcal G(n,p)$, where $p$ is the probability of having each edge present in the graph independently of the others, if you have $$ \binom{n}{2}p=m. $$ Contrary to the well known name as "the Erdős–Rényi model," the latter model was formulated by Gilbert in 1959, whereas Erdős and Rényi studied the one, which you seem to be asking for.
The results about this model can be found in most of the standard references, e.g., the treatise by Bollobas, but you can also start reading the original paper by Erdős and Rényi.