Can I open up my own business?

I would like to open my business for selling products through E-commerce channel but federal tax identification is needed in order to register LLC or C Corp business entities.

Will IRS allow me to get one if I only have work permit?

Yes , you can get EIN from IRS.

Yes....

Yes my father has his own business.

Yes you absolutely can. Why would you think otherwise? Now if you need some tax advice hit your boy up. You can operate an LLC as a sole proprietor as well in which case your SSN is sufficient.