Let's say we have already found a bijective homomorphism between two vector spaces or fields and thus know that they are isomorphic. Is it possible to find a homomorphism between them which is not bijective?
That is, if we find a non-bijective homomorphism between two vector spaces or fields are we allowed to conclude that they are not bijective?
Can a similar statement be made for other algebraic structures?
No. For vector spaces, you can always use the zero map. It is rarely a bijection, even when the two vector spaces are otherwise isomorphic. For fields, here is an explicit counterexample: $$f : \mathbb{R}(x) \to \mathbb{R}(x), \; P(x) \mapsto P(x^2).$$ Of course $\mathbb{R}(x)$ is isomorphic to itself, but $f$ is not a bijection because it isn't surjective ($x$ is not in the image).
There are some structures where, if you have two isomorphic objects, any morphism between the two is necessarily an isomorphism. For example, principal $G$-bundles over a given space. There are several examples there.