Short answer
There is at present no part of the Langlands programme – nor any other long–standing problem that the community had previously regarded as disconnected from logic and topos theory – whose solution relies in an essential way on the “topos-theoretic bridges’’ developed by Olivia Caramello. L. Lafforgue’s sentence you quote is therefore best read as an expression of hope rather than as a reference to existing achievements.

Below is some more detailed information (with references) so that you can see what has and has not been done.

1. What Caramello’s “bridges’’ are meant to do

• To every geometric first–order theory T one associates its classifying topos 𝔅(T).
• Two theories are Morita–equivalent if their classifying toposes are equivalent; in that case every piece of topos–theoretic information can be transported from one side to the other.
• The slogan “topos-theoretic bridges’’ means: choose a property P that can be expressed inside an arbitrary Grothendieck topos; translate P into the internal languages of the two theories; obtain two external theorems, one in each subject. Sometimes the translation of P is classical in one theory and non–trivial in the other, so a known theorem on one side yields a new theorem on the other.

All published work so far concerns theories that are already formulated in first-order geometric logic (Boolean algebras, distributive lattices, MV–algebras, ordered groups, profinite groups, Fraïssé classes, étale groupoids …). Automorphic representations, L-functions, Galois parameters, Shimura varieties etc. are not of this kind, so in order to treat Langlands material one would first have to recast it in a suitable geometric‐logical form, a task that has not been carried out.

2. Existing mathematical results that really use the “bridges’’

None of the following addresses the Langlands correspondence, but they are the papers in which Caramello’s methods do give new theorems whose statements themselves do not mention topoi.

(1) O. Caramello, “Fraïssé’s construction from a topos-theoretic perspective”, Log. Univers. 5 (2011) 127–139.
• Gives a unified proof of Fraïssé’s theorem and extends it from countable to arbitrary regular cardinals.
• Uses the Morita equivalence between the theory of the Fraïssé limit and the theory of certain ind-objects; the topos­-theoretic compactness criterion is the key new ingredient.

(2) O. Caramello, “A topos-theoretic approach to Stone-type dualities”, Trans. AMS 367 (2015) 3183–3221.
• Produces a general duality theorem that simultaneously recovers the classical Stone duality for Boolean algebras, Priestley duality for distributive lattices, and several new dualities (e.g. for MV-algebras).
• The bridge is the equivalence between the syntactic topos of the algebraic theory and the topos of sheaves on a certain topological category.

(3) O. Caramello, “Topological Galois theory”, Adv. Math. 222 (2009) 2107–2143 and the monograph “Topological Galois Theory” (Cambridge, 2017).
• Extends classical Galois correspondence far beyond the separable/normal field situation, covering for instance infinite Galois theory for (possibly non-normal) field extensions, covering theory of topological spaces, and pro-finite groupoids.
• The new correspondences are obtained by transporting knowledge along the Morita equivalence between atomic complete theories and continuous actions of their automorphism groups.

(4) O. Caramello, “Universal models and definability”, Math. Proc. Cambridge Philos. Soc. 154 (2013) 49–73.
• Gives an intrinsic characterisation of definability in a first-order theory and derives new preservation theorems for model-theoretic properties such as stability and superstability.

All of these papers end with theorems that can be (and in the articles are) stated in ordinary algebraic, topological or model-theoretic language with no mention of the ambient topos. The proofs, however, make essential use of the classifying-topos viewpoint.

3. Why there is no Langlands application so far

a) Expressibility. The objects central to Langlands – automorphic forms, perverse sheaves on stacks of shtukas, ℓ-adic Galois representations, local factors, trace formulas – live in derived or higher categories and depend on analytic conditions (growth, unitarity, harmonic analysis). None of this is currently encompassed by geometric first-order logic, so one does not (yet) have a classifying topos for “the theory of automorphic representations’’ in any reasonable sense.

b) Lack of candidate Morita equivalence. Even if one succeeded in giving a geometric‐logical axiomatization, one would still need a second geometric theory whose classifying topos is equivalent to it, and from which new information could be imported. No such pair of theories has been proposed.

c) Independent topos‐theoretic approaches already exist in the Langlands context – for instance Fargues–Fontaine’s work, Scholze’s pro-étale and diamonds, or V. Lafforgue’s stacks of shtukas – but these do not rely on Caramello’s transfer principle; they use the topos (or higher-topos) formalism in a more direct geometric way.

4. Bottom line

• L. Lafforgue’s remark is visionary rather than factual: it points out that the sought-for global Langlands correspondence would look very much like a Morita equivalence if one could cast it in the right language.
• Up to now nobody has succeeded in doing that, and there are no theorems in the Langlands programme whose proofs depend on Caramello’s techniques.
• Caramello’s “bridges’’ have, however, produced new results in several other areas (enumerated above), and these give a concrete idea of what the method can do when the hypotheses (geometric theories on both sides) are satisfied.

So the answer to your precise question is: no, there are at the moment no such papers; the connection with Langlands remains a speculative aspiration.