Short answer
qgis.utils.iface.activeLayer().crs().authid()
# returns: 'EPSG:26913'
Explanation
qgis.utils.iface.activeLayer()
returns a reference to the active QgsMapLayer.
QgsMapLayer.crs()
returns the crs or QgsCoordinateReferenceSystem for the layer.
QgsCoordinateReferenceSystem.authid()
returns the Authority identifier for the crs as a String.
However, this is assuming there is an active layer, it is of a vector type, and it has a valid crs. You will want to test for validity of those items before committing to reprojecting a raster.
If you are reprojecting, using GDAL.Dataset.SetProjection() will not suffice, since it will only assign a projection and not reproject (warp) the raster to the same as your vector layer.