There is a notion of vector-valued $L^p$-spaces. The typical definition is the following: Denote by $S(F)$ the space of (up to equality almost everywhere) the functions of the form $$ s(t) = \sum_{i=1}^{n} \chi_{A_i}(t) y_i$$ where the $A_i$ are pairwise disjoint measurable sets and such that for every continuous seminorm $q$ the sum $\sum_{i=1}^{m} \mu(A_i)^p q(y_i)^p$ is finite. We equip this space with the topology generated by the seminorms $$\pi_q(s) = \Big(\int_{X} q(s(t))^p\,\mathrm{d}\mu\Big)^{1/p} = \Big(\sum_{i=1}^{m} \mu(A_i)^p q(y_i)^p\Big)^{1/p}.$$ The space $L_{X,\mu}^{p}\{F\}$ is defined as the completion of $S(F)$.
This definition can be found in the literature at the following places:
- Chapter 46.1 in F. Trèves: Topological Vector Spaces, Distributions and Kernels
- These spaces are also mentioned in Grothendieck's thesis "Produits tensoriels topologiques et espaces nucléaires", see also Exemple 1 in his "Résumé des résultats essentiels dans la théorie des produits tensoriels topologiques et des espaces nucléaires" Here he refers to the Bourbaki book on integration but I could not find the general case there.
Unfortunately, it seems not much is known about these spaces. In the general case, it might even happen that elements of these spaces need not be functions with values in $F$ (see the quote in my question).