If the NP pays you more than $600 in a year, they issue you a 1099. Completely separate, if you hire a subcontractor (for any duties) and pay them more than $600 in a year, you issue them a 1099-NEC.
Sure, you can work for free. You can't take a tax deduction for the value of your time, unless you also report equivalent taxable income. If you have other business expenses like travel or supplies, you don't take a special charity deduction for that either. You just deduct the expenses in the normal way, and the fact that you don't get paid (and therefore don't have a profit) is where your tax "reduction" comes from.
