I feel silly for asking such a simple conceptual question, but it's something that's been bugging me for a while.
Is there any notation that makes it possible to define a piecewise equation that contains one or more vertical lines? For instance, how would one attribute two lines $y=2x+1$ (for $-1<x<3$) and $x=3$ to a single piecewise-defined "function" $f$?
Yes and no, I noticed you put function in scare quotes, which was smart since any graph with two points, one above the other will not be a function of $x$. In this particular case, you might be able to write part of this set with $x$ as a function of $y$, something like
$$ x(y) = \begin{cases} 3 & y\ge 7 \\ \frac{y-1}{2} & -1 < y < 7.\end{cases}$$
Note this only includes a portion of the vertical line in order to make it a proper function.
If you want the whole set I'd just do the following:
$$ \{ (x,y) \in \Bbb{R}^2 : (-1 < x <3 \text{ and } y=2x+1) \text{ or } x=3\}$$
Since it's no longer a function, just write it as a set.