Do vectors need to work in translated coordinate systems as well as rotated coordinate systems? In the Feynman Lectures on Physics, Chapter 11, section 5 (http://www.feynmanlectures.caltech.edu/I_11.html#Ch11-S5) Feynman states: "In order for it to be a vector, not only must there be three numbers, but these must be associated with a coordinate system in such a way that if we turn the coordinate system, the three numbers “revolve” on each other, get “mixed up” in each other, by the precise laws we have already described."

Is there not an equivalent requirement for when we translate the coordinate system?

If we ignore y and z components and just focus on x components, so we can describe each point in our coordinate system using only one number, we have the points:

A(a), B(b), and C(a + b).

In a new coordinate system, translated by α to the right, we have the points:

A(a'), B(b'), C(a' + b').

Substituting into the equation linking the two coordinate systems (x' = x - α), we get:

a' = a - α, b' = b - α, a' + b' = a + b - α

However, it is clear that adding a' and b' gives a' + b' = a + b - 2α, not a' + b' = a + b - α - they don't match.

Is my reasoning here wrong or is there simply no requirement for translations of coordinate systems as there is with rotations?
In that context, that is as in linear algebra, vectors are considered "applied" to the origin, therefore we are not interested in translations. In particular the text is referring to a rotation around the $z$ axis.
In that context, that is as in linear algebra, vectors are considered "applied" to the origin, therefore we are not interested in translations. In particular the text is referring to a rotation around the $z$ axis.