Are bundle gerbes bundles of algebras?  The category of line bundles (possibly with connection)
on a smooth manifold M can be defined in two different ways:
The first definition uses transition functions that satisfy a cocycle condition
(possibly with additional data of a 1-form that defines a connection),
and the second definition uses invertible vector bundles (possibly with connection),
where vector bundles are defined
algebraically as dualizable modules over the algebra of smooth functions
or geometrically as vector spaces in the category of smooth submersions over M
and connections are defined algebraically as certain linear maps satisfying the Leibniz identity,
geometrically in terms of subbundles, or topologically as certain functorial field theories.
The above two categories (with or without connection) are equivalent to each other:
There is a canonical fully faithful functor from the first category to the second one.
This functor is essentially surjective and hence an equivalence of categories,
even though the construction of an inverse functor requires a choice.
The first definition was categorified by Michael Murray in 1994,
the result being the bicategory of bundle gerbes (with or without connection).
Can one categorify the algebro-geometric definition of a line bundle
in such a way that there is an equivalence from Murray's bicategory of bundle gerbes
to the bicategory of categorified line bundles,
thus obtaining a “chart-free” definition of bundle gerbes?
Naïvely, one might expect that vector bundles should categorify to bundles of algebras.
Just as any fiber of a line bundle is noncanonically isomorphic to the vector space of scalars,
any fiber of a categorified line bundle should be
noncanonically Morita-equivalent to the algebra of scalars.
Furthermore, if a line bundle over M is equipped with a connection,
then any path in M gives an isomorphism between the corresponding fibers.
These isomorphisms can packaged in a 1-dimensional topological field theory,
and in fact a theorem by Florin Dumitrescu, Stephan Stolz, and Peter Teichner
shows that any 1-TFT comes from a vector bundle with connection,
thus giving an alternative definition of a vector bundle with connection.
Similarly, if a bundle gerbe over M is equipped with a connection,
then any path in M should give a Morita equivalence between the corresponding fibers.
Moreover, a bigon in M should give an isomorphism between the corresponding Morita equivalences.
Thus, one should be able to package the above parallel transports in a local 2-TFT.
Invertible morphisms between bundle gerbes should correspond to bundles of invertible bimodules
(i.e., Morita equivalences) and non-invertible morphisms (introduced by Konrad Waldorf in his paper
More Morphisms between Bundle Gerbes)
should correspond to bundles of non-invertible bimodules.
Several papers in the literature are closely related to the above question.
For example, Corollary 4.9 in the paper by Urs Schreiber and Konrad Waldorf
Connections on non-abelian gerbes and their holonomy
proves that the bicategory of bundle gerbes with connection over M
is equivalent to the bicategory of transport functors with values in some bicategory.
This statement would almost resolve the above question if not for the fact that
the target bicategory has only one object, which prevents one from considering
constructions like the algebra of global sections of a bundle gerbe
(the analogous construction for line bundles (the vector space of global sections)
plays an important rôle in geometric quantization and other areas of mathematics).
In fact, the algebra of global sections of a bundle gerbe is interesting enough
to warrant its own question:
Can one construct the algebra of global sections of a bundle gerbe
that categorifies the vector space of global sections of a line bundle?
Finally, I am also interested in the answers to the above questions
for the case of arbitrary vector bundles (respectively non-abelian bundle gerbes)
and not just line bundles.
 A: There is a bicategory of Dixmier-Douady bundles of algebras which is equivalent to the bicategory of bundle gerbes. In particular, sections into these bundles form algebras. 
The price you pay is that the bundles are infinite-dimensional; for that reson I am not sure if that picture persists in a setting "with connections".
I do not know a good source for the bicategory of Dixmier-Douady bundles or for the equivalence. Everything depends certainly on the type of morphisms you consider between the bundles; they clearly have to be of some Morita flavor. You may look into Meinrenken's "Twisted K-homology and group-valued moment maps", Section 2.1.1 and 2.1.4. In Section 2.4 Meinrenken indicates indirectly that his bicategory of Dixmier-Douady bundles is equivalent to the one of bundle gerbes, by transfering the notion of a multiplicative bundle gerbe (which depends on the definitions of 1-morphisms and 2-morphisms) into his language. 
Side remark: a bundle gerbe is not the direct generalization of transition functions of a bundle. There is one step in between, namely a bundle 0-gerbe: instead of open sets, it allows for a general surjective submersion as the support for its transition functions. If you take bundle 0-gerbes instead of transition functions, the functor you mentioned at the beginning of your question has as canonical inverse functor. See my paper with Thomas Nikolaus "Four equivalent versions of non-abelian gerbes".
Added (after thinking a bit more about the question): If you want to categorify the vector space of sections into a vector bundle, you first have to fix a categorification of a vector space. An algebra is one possible version of a "2-vector space", probably due to Lurie. Another version, due to Kapranov-Voevodsky, is to define a 2-vector space as a module category over the monoidal category of vector spaces (add some adjectives if you like).
Let us define a section of a bundle gerbe $\mathcal{G}$ over $M$ to be a 1-morphism $s: \mathcal{I} \to \mathcal{G}$, where $\mathcal{I}$ is the trivial bundle gerbe. Then, sections form a category, namely the Hom-category $Hom(\mathcal{I},\mathcal{G})$ of the bicategory of bundle gerbes (the one with the "more morphisms" defined in my paper which was mentioned in the question). 
The category $Hom(\mathcal{I},\mathcal{G})$ of sections of $\mathcal{G}$ has naturally the structure of a module category over the monoidal category of vector bundles over $M$. Indeed, a vector bundle is the same as a 1-morphism between trivial gerbes, i.e. an object in $Hom(\mathcal{I},\mathcal{I})$. Under this identification, the module structure is given by composition:
$$
Hom(\mathcal{I},\mathcal{G}) \times Hom(\mathcal{I},\mathcal{I}) \to Hom(\mathcal{I},\mathcal{G}).
$$
The functor which regards a vector space as a trivial vector bundle induces the claimed module structure over vector spaces.

Summarizing, sections of bundle gerbes do not directly form algebras, but they form Kapranov-Voevodsky 2-vector spaces.

