Difficulty understanding the variational bicomplex Let $E\rightarrow X$ be a smooth fiber bundle, with (infinite-order) jet bundle $J_\infty\rightarrow X$.
I am reading about the variational bicomplex; as stated at that nLab link, it is a bicomplex structure on the de Rham complex of $J_\infty$ (regarded as an infinite-dimensional manifold).
The expression $\Omega^n(J_\infty) = \bigoplus_{h+v=n} \Omega^{h,v}$ at that link suggests that the variational bicomplex comes from some direct-sum decomposition $\Omega^1(J_\infty) = A\oplus B$ where $A,B$ respectively would be "horizontal and vertical" 1-forms, and correspond in the aforementioned notation to $(h,v)$ being $(1,0),(0,1)$ respectively.
I understand that we could take e.g. the "horizontal" 1-forms on $J_\infty$ to be the ones which vanish on $\mathrm{ker}(d\pi)$, where $\pi : J_\infty \rightarrow X$ is the projection.
However, I don't see how a choice of "vertical" 1-forms on $J_\infty$ arises. Wouldn't this be equivalent to a choice of connection on the smooth bundle $J_\infty\rightarrow X$?

Edit: Originally I asked about horizontal/vertical vector fields; instead I've changed the question to be about horizontal/vertical 1-forms.
 A: I was going to add a comment but I just made an account; so I cannot yet comment.
For a bundle $E \rightarrow X$, the space of vertical differential forms $\Omega^*_V(E)$ is defined to be the quotient of the de Rham complex $\Omega^*(E)$ by the differential ideal of horizontal forms $\Omega^*_H(E)$ which, as you noted, is canonically defined. A connection allows you to choose representatives in this quotient.
The above is for a general bundle $E$, so there is no canonical choice of representatives for the space of vertical differential forms; i.e., there is no canonical choice for $\Omega^*_V(E)$ as a subalgebra of $\Omega^*(E)$ instead of a quotient. However, as @Alex K mentioned, for the case where $E$ is the infinite jet bundle, there is a canonical flat connection.
Another reference: for the explicit construction of vertical forms in the variational bicomplex, see Anderson, The Variational Bicomplex (pages 15 to 18 in the document's numbering, or pages 44 to 47 of the pdf). I like this exposition because it shows how the vertical and horizontal exterior derivatives arise explicitly.
