Proposition 3.18 of Voisin's Hodge Theory and Complex Algebraic Geometry I says that, if $X$ is compact Kahler and $E$ is a holomorphic vector bundle over $X$, then $\mathbb{P}(E)$ is Kahler. Since $E$ embeds as an open submanifold of $\mathbb{P}(E \oplus \mathbb{C})$, this establishes your result for $X$ compact, and I think her proof could be simplified if you just want tthe vector bundle version and not the projective bundle version. But it looks to me like she actually is using compactness in a nontrivial way.

Proposition 3.18 of Voisin's Hodge Theory and Complex Algebraic Geometry I says that, if $X$ is compact Kahler and $E$ is a holomorphic vector bundle over $X$, then $\mathbb{P}(E)$ is Kahler. Since $E$ embeds as an open submanifold of $\mathbb{P}(E \oplus \mathbb{C})$, this establishes your result for $X$ compact, and I think her proof could be simplified if you just want the vector bundle version and not the projective bundle version. But it looks to me like she actually is using compactness in a nontrivial way.