Name of the following structure? I need the name of the algebraic structure that is like a vector space, but the vectors form a monoid, not a group; the field and the scalar multiplication stays the same.

I think the closest thing to what you are talking about is a semimodule over a semifield which is a commutative monoid acted upon by a semifield. For both the semimodule and the semiring, we've dropped the condition that what used to be an abelian group now does not necessarily have inverses for its elements.
The situation is like Joppy mentioned in the comments: if you $V$ which is a vector space except that it does not have additive inverses, and you have a field acting on it in the normal way, then in fact $-1\cdot v$ is defined for every $v\in V$, and that is the additive inverse of $v$. So to have something truly different you need to drop the requirement of additive inverses from "the field" as well.
Actually everything above can still be said if we're talking about just a semimodule over a semiring (with identity.)
