What is the max current I could draw from a 9V battery?

Here http://data.energizer.com/PDFs/522.pdf is the datasheet for the Energizer alkaline 9V battery. Here http://data.energizer.com/PDFs/la522.pdf is the one for the lithium version. You can get the answer to your question from these sheets.

For 150mA @ 9V, consider using 6xAA cells instead.