Does the solar wind correspond to a net electrical current? The solar wind is a stream of ions emitted from the sun. Are there more positive than negative charges released? That is, is there any imbalance in the charge emitted from the sun in the solar wind?
 A: 
Are there more positive than negative charges released?

In general no, and this statement by itself would imply you were asking about whether the sun charges up, not about current.
I wrote a detailed answer at https://physics.stackexchange.com/a/253491/59023 that explains why stars remain mostly neutral and that most observations [e.g., Bale et al., 2013; Pulupa et al., 2014] support the idea that the solar wind, on the macroscopic scale, has no net current flowing out from the sun.

That is, is there any imbalance in the charge emitted from the sun in the solar wind?

Since plasmas are made of roughly equal positive and negative charges and that electric fields typically act to eliminate themselves, the short and simple answer is no.  If the sun were to charge up one could imagine the resulting net electric field would act to stop the solar wind from flowing.  Indeed it is the case that such static models predict solar breezes instead of the observed supersonic solar wind.
In summary, the most recently accepted models (which are supported by observations), called exospheric models, argue in favor of no net macroscopic current and insignificant net charging of the star.
Update
A recent paper by Wilson et al. [2019] directly calculates the net current density (normalized to the core electron thermal current density) of the electrons in the ion rest frame finding most time periods are within a few percent of zero.  Note that $\mathbf{j}_{e} = -e \ \sum_{s} \ n_{s} \ \mathbf{v}_{s} = 0$ was not a constraint imposed during the fitting process of the velocity distribution components.  That is, the small magnitude of the current density in the ion rest frame is a consequence of the solar wind actually maintaining $\lvert \mathbf{j}_{e} \rvert$ ~ 0 in the ion rest frame for most intervals.  The deviations from this are associated with changes in the magnetic field magnitude and direction, as one would expect from Ampere's law, e.g., at the shock ramps examined in this work.
Similar results were much more recently observed using data from the Parker Solar Probe mission [e.g., Halekas et al., 2019 (arXiv eprint 1912.02216v1, to be published in ApJ in Feb. of 2020)].  The data shows that this constraint appears to hold even at smaller solar radial distances, down to below ~0.3 AU.
A: The sun is a ball of plasma. It is essentially a group of electrons and nuclei whizzing around. These electrons and nuclei correspond to no net charge. The get flung out into space still in the form of plasma. There is still no net charge. The individual particles have charge but no net charge.
