You can use virtual QAction from QgsInterface Class. There is for your purpose,
actionSplitFeature() that you can trigger when the user click on your button.
for example you can define a function like that (you can add the code you need for your task, like set the layer to work with, selecting features,...):
and connect your button to this function like that:
so when you click your button, the split features tool will be trigger.
Have a look to the QgsInterface class documentation to see all Qaction available : https://qgis.org/api/classQgisInterface.html