If I factor out a matrix from an equation and then am left with 0 on one side, can I effectively cancel out the vector by dividing both sides by it?
A trivial example being $$ \begin{align} AB\hat{x} - AC &= 0 \implies\\ A(B\hat{x} - C) &= 0 \implies\\ B\hat{x} - C &= 0 \end{align} $$
while also obviously considering that $A=0$ is a possibility or has been assumed to not be $0$
On
Depending on the dimensions of the various things, which you do not define but I imagine $A,B,C$ to be square or rectangular matrices and $X$ to be a column vector, your equation does not make sense as it sums matrices of different sizes.
Even if the equation would make sense the answer would be no as you can simplify $A$ only if it is invertible (so you are actually multiplying by $A^{-1}$). In general with matrices you can have $AB=0$ with $A\neq 0$, $B\neq 0$.
The correct analogy to the usual $xy=0$ implying $x=0$ or $y=0$ over the reals (or really any integral domain) uses invertibility of square matrices:
When $A$ is a (real or complex) square matrix and $x$ is a vector, the equation $Ax=0$ implies that $x=0$ or $A$ is not invertible (which allows more than just $A=0$).
Hence, the implication $$Ax=0\ \Longrightarrow\ x=0$$ is only true, when $A$ is an invertible matrix.