This is basically a reference request, though I would also love to see the idea of a proof.
Suppose we are given a smooth Hilbert bundle $\pi: E \to X$ over a $n$-dimensional smooth manifold $X$ where the fibres are infinite-dimensional separable Hilbert spaces. There are some subtleties here since one needs to carefully define what is meant by smoothness and local trivialisations in infinitely many dimensions. However, since I have not managed to find a good reference, I do not really care about these details at this point.
I have read multiple times that there always exists a global orthonormal frame thanks to the infinitely many dimensions, i.e a global section of the corresponding orthonormal frame bundle. But I am struggling to find any proofs of this fact, let alone a context with rigorous definitions.
The key point here is Kuiper's theorem which says that if $H$ is an infinite dim. real or complex separable Hilbert space then $GL(H)$ is contractible in the norm topology.
The conclusion then follows from the theory of classifying spaces for (principal) bundles. Which tells you that $GL(H)$-bundles over $M$ up to isomorphism are in bijection with homotopy classes of maps $M \to B(GL(H))$ where $B(GL(H))$ is some topological space.
Since the inclusion $\{1\}\to GL(H)$ is a group homomorphism and by Kuiper's theorem is a homotopy equivalence then by this answer we see that $B(GL(H)) $ is homotopy equivalent to the classifying space of the trivial group which can be taken to be a point. Consequently there is only one homotopy class of maps $M \to B(GL(H))$ and thus only one iso. class of bundles (the trivial one).
A good reference are these notes by Dan Freed https://web.ma.utexas.edu/users/dafr/M392C-2015/Notes/lecture12.pdf
----Another possible approach---- Instead of using classifying spaces we can use obstruction theory. We want to construct a global section of a bundle $GL(H)\to P\to M$, we give $M$ the structure of a CW-complex, we choose a section over the $1$-skeleton and we try to extend it up to the $\mathrm{dim} M$-skeleton, The obstructions to do so lie in $H^i(M, \pi_{i-1}GL(H))$, $i\geq 2$ but since by Kuiper $GL(H)$ is homotopy equivalent to a point $\pi_{i-1}GL(H) = (0)$ consequently all the obstruction vanish and we can construct a global section.