Integration in a finite dimensional vector space

If you do not want to use coordinate systems, a more intrinsic way to reduce to scalar functions is defining $\int_Gf(g) \mu(dg)$ as the (unique) element $v\in V$ such that for any linear form $\phi\in V^*$ one has $\int_G \langle \phi,f(g)\rangle\mu(dg)=\langle \phi,v\rangle$.

Definitely do not pick a basis. You also don't have to pick an inner product on $V$ or reduce to scalar-valued functions.

You can define integration with respect to a measure in direct way as a limit without having to do any kind of decomposition into real or imaginary parts or positive or negative parts as is often presented for integration of real or complex valued functions. See the treatment of integration in Lang's "Real and Functional Analysis", where the functions being integrated take values in a Banach space. That applies in particular to functions with the values in a finite-dimensional complex vector space $V$ (which has a canonical Hausdorff topology making it complete for all vector space norms on $V$).