If you have a non-profit organization can you do fundraising?

Absolutely! That's how non-profits survive. School PTO's and PTA's are non-profits and hold fundraisers all school year. There is a limit to how much you can earn though before it's considered income. I'm not sure but it probably depends on type of or purpose of non-profit. The limit for PTO's was $25000 per year a couple of years ago.