Suppose we know the components of the vector $\vec{c}$ which satisfies
$c_i=\sum_{j,k,...p}C_{i,j,k,...p}A_jB_k....X_p$
where I don't know the tensor $C$ but I know the vectors $A,B,...X$.
Can I get the new vector $\vec{d}$ which satisfies the same equation but with the vector $X$ in the outer product $A\otimes B\otimes...X$ replaced by another vector $Y$?
View $\sum_{j,k,\dots}C_{i,j,k,\dots,p}A_jB_k\dots$ as a single object $D_{i,p}$. You know $\sum_pD_{i,p}X_p$ and you want to know $\sum_pD_{i,p}Y_p$. This is impossible. Clearly knowing how a matrix acts on one vector isn't enough to tell you how it works on all vectors. However if you knew $\sum_pD_{i,p}X_p$ for enough different values of $X$ then you might be able to write $Y$ as a linear combination of the different $X$s, at which point you could find $\vec{d}$.