With AC, it is easy to see that any vector space is injective, and free, therefore alse flat and projective. Without AC, vector spaces can be not free. Are they must be projective modules? Flat modules? What about injectiveness?

I think vector spaces must still be flat (tensor product is exact). I don't think any of the steps in the following proof use choice, although it's quite possible I'm mistaken: Finite dimensional vector spaces are flat. Every vector space is the filtered colimit of its finite dimensional subspaces. Tensor products commute with colimits. Filtered colimits of exact sequences of vector spaces are exact. 


(I completely revamped my answer, the previous version (link) had a consistency result with a particular example, this feels much better as it establishes a full equivalence result instead.) The answer is no. For projectivity and injectivity. The reason is that the axiom of choice is equivalent to the following statement:
If every vector space is projective, consider the case that $V$ is a vector space, $K$ is a subspace and $T\colon V\to V/K$ is the quotient map. Then there is some $h\colon V/K\to V$ such that $h\circ T$ is the identity function. In particular, this means that $K+\operatorname{im}(h)=V$, but $h$ cannot map any nonzero vector into $K$ so this is a direct sum indeed. Therefore the statement "Every vector space is projective" implies the axiom of choice. Similarly for injectivity, take the identity function $K\to V$, then there is a projection map whose kernel is a direct complement for $K$. I think that a relevant paper to mention here would be the following:


