Let's say I have a vector a.
I would like to construct a matrix or vector b such that if I multiply a and b, I get the absolute value of a.
In other words I would like to calculate the absolute value of a vector by multiplying it with a matrix or a vector.
Is such a matrix/vector possible to construct, assuming that I do not know anything about a?
I would like to do this without using a.
To answer your question of if there is a single $b$ which works for any vector $a$ where $b$ does not depend on $a$, first note that projections are self-adjoint operators, that is $\Pi = \Pi^*$.
Define $\Pi_a$ to be the projection onto the subspace generated by the vector $a$.
$|a| = \langle a, b\rangle = \langle \Pi_a a, b\rangle = \langle a, \Pi_a b\rangle$
also, $\langle a, b\rangle = |a||b|\cos \theta\leq |a|$
These together implies that $\Pi_a b = b$, which is to say that $b\in \text{span}\{a\}$
Take some vector $c$ such that $c\perp a$ and $|c|\neq 0$
Then $\langle c, b \rangle = \langle c, \Pi_a b\rangle = \langle \Pi_a c, b\rangle = \langle 0, b\rangle = 0 \neq |c|$
This shows that there cannot exist a $b$ in any space that includes perpendicular vectors, but still could exist in 1-deminsional space. For one-dimensional space, the number $1$ does the trick if only using positive numbers, but including negative numbers will ruin that as well.
tldr: $b$ will always depend on what $a$ is.