I have a working stand-alone pyqgis application that can display vector and raster layers, the question is how to access the layer style manager as in QGIS's layer property, where users can change the symbology of a layer? I am hoping there is a class already defined in the QGIS core library that I can just call from my application.
Yes, that is possible! You can use the QgsSymbolV2SelectorDialog class to do that. Here a minimal example form PyQGIS APIs on how the main class constructor works:
SymbolSelectorDialog = QgsSymbolV2SelectorDialog( layer.rendererV2().symbol(), QgsStyleV2.defaultStyle(), layer, None, False)
layeris the reference to
QgsVectorLayer. Please, take a look at the documentation!