The definition of vector space is always defined over a field. Even for finite fields (I believe, for example $\mathbb{Z}\mathrm{mod}p$).
Integers are not a field because there is no general inverse element, yet I can see that tuples of integers with usual element by element sum, seem to form a vector space fulfilling all (?) the axioms. https://en.wikipedia.org/wiki/Vector_space#Definition
Is there an implicit division (of vectors) requirement that integers cannot fulfill?
In any case, is there a name for a "vector space with integers"? or is it just not interesting enough? in the same way that "integer" complex numbers are called Gauss integers.
NOTE: After posting this question I got referenced to this other related but more generic question: Does the "field" over which a vector space is defined have to be a Field?
I guess it was not about the expressions of the axioms (and those of field implicitly) but what can be proved with these axioms (e.g. that the cardinal of any basis is the dimension).