Maybe more a suggestion than an answer.
You could replace the space of all maps $(\mathbb{R}^k)^{\mathbb{R}^d}$ with the space of all continuous maps $C:=C(\mathbb{R}^d,\mathbb{R}^k)$. We can then endow $C$ with the topology of uniform convergence on compacta (aka compact convergence), which coincides in this case with the compact-open topology. According to mathoverflow this topology on $C$ is metrizable, and thus completely regular (see wikipedia).
On the topological space $C$ it is natural to consider $\tau$-additive probability measures and the narrow convergence of measures.
Note that, according to Fremlin Cor. 437L, narrow convergence and the vague convergence coincides for $\tau$-additive probability measures on completely regular topological spaces. So for $C$ there is no ambiguity here.
Then you could model $F$ as a measurable map $F: \Omega \to C$, where $C$ carries the Borel $\sigma$-algebra w.r.t. to the mentioned topology, such that the distribution $\mu$ of $F$ on $C$ is $\tau$-additive.
A sequence $(\mu_n)_{n \in \mathbb{N}}$ of $\tau$-additive probability measures on $C$ convergences to a $\tau$-additive probability measure $\mu$ on $C$ narrowly/weakly/vaguely if and only if for all bounded continuous functions $g: C \to \mathbb{R}$ we have for $n \to \infty$ the convergence: $$ \int g \,d\mu_n \to \int g\,d\mu.$$
Please carefully check for yourself if all arguments made are correct.