I would bet it is the battery though. These alternators on the Toyota's (Denso) seem to last forever. On my 83 Toy 4X4 when I got rid of it around 2005, I had the factory alternator on it still, 22 years old and working like a champ. Go to Autozone if there is one around you and they will check the battery for free for you and let you know if the alternator is charging. If not one close, you can buy a cheap voltmeter from Walmart, etc for around $20. It should be charging somewhere around the neighborhood of 14 volts or more at the battery when running.
My 2 cents. Good luck,