In Atiyah's K-theory book, he gives a proof of the Tietze extension theorem to vector bundles. He gives the proof only for compact Hausdorff spaces, but I want to generalise it to paracompact Hausdorff spaces. The statement is the following:
$\textbf{Lemma}$ : Let $B$ be paracompact Hausdorff and $p:E\to B$ a vector bundle. Let $A\subset B$ closed. Then a section $s:A\to E|A$ can be extended to a section $s:B\to E$.
$\textit{Proof}$: Let $s:A\to E|A$ be a section. Then for every $x\in B$ we can extend the section to an open set $U_x$ containing $x$. If $B$ were compact, then we could cover $B$ with these sets $U_x$ and take a finite subcover. However, since it is only paracompact, the best we can do is take a locally finite refinement $\{V_\alpha\}$. The objective would be to have sections from this refinement and glue them together via partitions of unity. Clearly this doesn't necessarily have to work for paracompact spaces since the indexing set of $\{V_\alpha\}$ need not be finite so the partitions of unity need not converge.
There is a lemma that I feel should help here that says: for every open cover, there is a countable open cover such that every element in the second cover is a disjoint union of open sets, each of which is contained in elements of the original cover.
There is no "convergence" issue at all. That's the entire point of having a partition of unity subordinate to a locally finite open cover: near any point, only finitely many terms of the partition of unity are nonzero, so you just have a finite sum (which is furthermore guaranteed to be continuous since it is a sum of continuous functions). So, you pick your partition of unity to be subordinate to $\{V_\alpha\}$, and there will be no issue adding everything up.