Unless I have not botched things horribly, here is a general proof that cosheafification exists following the outline J. Curry gave.

Fix a target category $\mathcal{A}$ that is complete, cocomplete and locally presentable. Examples include categories of algebraic objects like linear spaces, categories of sheaves, the category of Banach spaces and linear contractions, etc.

Let $\Omega$ be a small site and $\mathbf{PCoShv}(\Omega, \mathcal{A})$ the category of precosheaves, that is, functors $\Omega\longrightarrow \mathcal{A}$. The category $\mathbf{PCoShv}(\Omega, \mathcal{A})$ is complete and cocomplete with limits and colimits computed pointwise and locally presentable by [AR, corollary 1.54].

Cosheaves are defined dually to sheaves, that is, they send the cone induced by a sieve of the small site to a colimiting cone. Denote the full subcategory of cosheaves by $\mathbf{CoShv}(\Omega, \mathcal{A})$. By the interchange of colimits theorem, $\mathbf{CoShv}(\Omega, \mathcal{A})$ is cocomplete and the inclusion functor $\mathbf{CoShv}(\Omega, \mathcal{A})\longrightarrow \mathbf{PCoShv}(\Omega, \mathcal{A})$ is cocontinuous.

proposition: $\mathbf{CoShv}(\Omega, \mathcal{A})$ is accessible.
proof: The category of sheaves on a small site is limit-sketchable by [RP, pg. 331]. The same construction yields by duality that the category of cosheaves is the category of models of a colimit sketch, therefore it is accessible by [AR, corollary 2.61]. Q. E. D.

proposition: $\mathbf{CoShv}(\Omega, \mathcal{A})$ is complete and locally-presentable.
proof: we already know that $\mathbf{CoShv}(\Omega, \mathcal{A})$ is cocomplete and accessible, so the result follows from [AR, corollary 2.47]. Q. E. D.

theorem: the full inclusion $\mathbf{CoShv}(\Omega, \mathcal{A})\longrightarrow \mathbf{PCoShv}(\Omega, \mathcal{A})$ has a right adjoint.
proof: follows from the previous results and the fact that cocontinuous functors between locally presentable categories have right adjoints. This adjoint functor theorem can be pieced together via the concept of totality. More precisely, locally presentable categories are total by [MK, corollary 6.5 and remark 6.6] and total categories are compact, that is, every cocontinuous functor has a right adjoint -- this is [MK, theorem 5.6].

Bibliography:
[AR] J. Adamek, J. Rosicky - Locally presentable and accessible categories, Cambridge University Press (1994).

[MK] Max Kelly - A survey of totality for enriched and ordinary categories, Cahiers de Top. et Géom. Diff. Catégoriques, 27 no. 2 (1986), p. 109-132

[RP] R. Pare - Some applications of categorical model theory, in Categories in Computer Science and Logic, Contemporary Mathematics, vol. 92 (1989)

edit: cleaned up, added a couple of references and made mention of the correct adjoint functor theorem.