Yes you can do your taxes without a 1099 form although it's best to use the 1099 form when filing the taxes to get the best accuracy for your income.
If you over report the income through the IRS without a 1099 then you won't be dinged or be in trouble for it.
I once over reported my income to the IRS because I didn't have the 1099 and I had to pay more taxes but when I did file with a 1099 the next year then I got a refund.
If you work a job with an employer they should give you a 1099 IRS tax form to fill out at the end of the year and if not you can ask the employer for a 1099 tax form.
If you work at home or for yourself you can go online and print out a free 1099 tax form to mail to the IRS.
I make some income through Adsense and blogging and they send me a 1099 tax form at the end of the year to fill out so that is one way I can report the Adsense income to the IRS.
But if you work for yourself then you will need to print out a 1099 form but you're not required to use one but it makes reporting the income easier and more accurate.