Is a coordinate system a requirement for a vector space? In other words, can a vector space exist, and not have a coordinate system?
I'm asking because in the definitions that I've seen of a vector space, there's no mention of a coordinate system.

For any finite dimensional vector space $V$, there exists a coordinate system (it is not unique). Indeed, let $ n := \dim V$. Then you can find a linearly independent set $\mathcal{B} = \{u_1, \dots, u_n\}$ of $n$ vectors that will generate $V$, and $\mathcal{B}$ can be taken as a coordinate system if your order its elements.
However, the vector space structure doesn't include one particular specified coordinate system. In other words, to define a finite dimensional vector space, you don't need to specify a coordinate system, but you can always find one if you want.
In the general case of (maybe infinite) vector space, the existence of a coordinate system is not a requirement, and there are examples of vector spaces with no coordinate system, as @Hagen von Eitzen pointed out in a comment.
