If I do contract work for an organization and receive a 1099 NEC, but can't write anything off as a business expense, do I need to fill out a Schedule C?

Yes, you will still need to report it on Schedule C as this is self-employment income and subject to Self-Employment taxes.

 

 Whether or not you have expenses does not change how the income is treated, the expenses just lower the amount that is taxed.  You are correct, if you don't have a space set up that you use exclusively for business, then you cannot deduct a home office.  However, if you can determine the portion of your internet that is used for business, then you can deduct that amount.  So if you use the wifi 20% for business, then you would be able to deduct 20% of the bill.