Yes, L<sub>p</sub>-spaces can be defined for arbitrary hermitian vector bundles.

For the sake of convenience I denote L<sub>p</sub>=L<sup>1/p</sup> (see [this answer](https://mathoverflow.net/questions/28147/why-do-we-care-about-lp-spaces-besides-p-1-p-2-and-p-infinity/28169#28169) for a motivation), in particular L<sub>0</sub>=L<sup>∞</sup> and L<sub>1/2</sub>=L<sup>2</sup>.
As explained in the link above, p is an arbitrary complex number such that ℜp≥0.

Suppose M is an arbitrary smooth manifold, possibly non-compact,
and V is a finite-dimensional hermitian vector bundle over M.
Let me stress that we do not need any additional data on M
such as a metric, a volume form, a density, or an orientation.

Recall the definition of the line bundle Dens<sub>p</sub>(M) of p-densities on M for an arbitrary
complex number p (no restrictions on the real part of p):
Every fiber of Dens<sub>p</sub>(M) is the vector space of all set-theoretical maps f: Λ<sup>top</sup>(TM) \ {0} → **C**
such that for all λ∈**C** \ {0} and for all x∈Λ<sup>top</sup>(TM) \ {0} we have f(λx)=|λ|<sup>p</sup>f(x).  In particular, Dens<sub>1</sub>(M) is the tensor product of the line
bundle of top-degree differential forms on M and the line bundle of orientations of M.

Note that for all p∈**R** the line bundle Dens<sub>p</sub>(M) has a canonical orientation,
in particular it is trivaliazable.
Moreover, for all p∈**C** \ **R** the line bundle Dens<sub>p</sub>(M) is also trivializable,
even though it does not possess a canonical orientation.
However, only Dens<sub>0</sub>(M) has a *canonical* trivialization.

Observe that all bundles Dens<sub>p</sub> combine together in a **C**-graded unital \*-algebra,
i.e., we have the unit **C**→Dens<sub>0</sub>(M), the multiplication Dens<sub>p</sub>(M)⊗Dens<sub>q</sub>(M)→Dens<sub>p+q</sub>(M), and the involution (Dens<sub>p</sub>)\*→Dens<sub>p\*</sub> (the first star denotes the conjugation of the complex structure on a vector bundle, the second star denotes the conjugation of complex numbers).
All these morphisms are isomorphisms of line bundles.

For any t∈**C** and any p>0 we also have the power operation
Dens<sub>p</sub><sup>+</sup>(M)→Dens<sub>tp</sub>(M).  This is *not* a morphism of vector bundles,
because it is non-linear for t≠1 and Dens<sub>p</sub><sup>+</sup>(M)
is a fiber bundle, not a vector bundle.
However, the power operation is still a morphism of fiber bundles,
in particular we can talk about powers of positive sections of Dens<sub>p</sub>(M).

We have a canonical integration map ∫: C<sup>∞</sup>(Dens<sub>1</sub>(M))→**C**.
We use this map to define norms on spaces C<sup>∞</sup>(Dens<sub>p</sub>(M)) for all p∈**C** such that ℜp>0.
First we send f∈C<sup>∞</sup>(Dens<sub>p</sub>(M)) to |f|=(f\*f)<sup>1/2</sup>∈C<sup>∞</sup>(Dens<sub>ℜp</sub>(M)).
Observe that f\*f and |f| are positive with respect to the canonical orientations on Dens<sub>2ℜp</sub>(M) and Dens<sub>ℜp</sub>(M).
Then |f|<sup>1/ℜp</sup>∈C<sup>∞</sup>(Dens<sub>1</sub><sup>+</sup>(M))
and we set ‖f‖:=∫⁠(|f|<sup>1/ℜp</sup>).  This is a norm for ℜp≤1 and a quasi-norm for ℜp>1.

The (quasi-)Banach space L<sub>p</sub>(M) is
the completion of C<sup>∞</sup>(Dens<sub>p</sub>(M)) in this (quasi-)norm.


If ℜp=0, then we complete C<sup>∞</sup>(Dens<sub>p</sub>(M))
in the weak topology induced by C<sup>∞</sup>(Dens<sub>1−p</sub>(M))
and obtain the Banach space L<sub>p</sub>(M).
(The norm of f∈C<sup>∞</sup>(Dens<sub>p</sub>(M)) can be defined
as the supremum of |f|∈C<sup>∞</sup>(Dens<sub>0</sub>(M))=C<sup>∞</sup>(M),
however, C<sup>∞</sup>(Dens<sub>p</sub>(M)) is not dense in L<sub>p</sub>(M)
in the norm topology.)

Thus we defined L<sub>p</sub>-spaces of the trivial line bundle on M for an arbitrary p∈**C**
such that ℜp≥0.
To extend this definition to an arbitrary hermitian vector bundle V
we replace f\*f by μ(f\*f) in the above definition of norm.
Here μ denotes the hermitan pairing on V.

All the usual theorems of measure theory like Radon-Nikodym, Riesz, Fubini, Tonelli
etc. hold in this more general setting.