Is the category of operator algebras over a field $K$ (that is, $K$-algebras that are the endomorphism algebra of some vector space) equivalent to the category of vector spaces over the same field $K$?
And if so, is there some general theorem about when this is true for the arbitrary category? Since the operator algebra is just the endomorphisms for each vector space.
No, these categories are very far from being equivalent. There is a correspondence between their objects but the morphisms are totally different. For instance, the category of operator algebras has an object $K$ (the endomorphisms of a 1-dimensional vector space) which has exactly two endomorphisms (the identity and the zero map). The category of vector spaces has no such object unless $K=\mathbb{F}_2$. (In the case $K=\mathbb{F}_2$, you can say there are exactly four morphisms from the vector space $K$ to the vector space $K^2$, whereas there is no operator algebra with exactly four morphisms from $K$, where in both categories $K$ is characterized as an object with exactly two endomorphisms.)