If the people who will be filling out the forms are using Adobe Reader, you will have to use Adobe Acrobat to enable Reader users to save the PDF (unless they are using Adobe Reader XI or DC, in which case they should have that ability already). Enabling Reader abilities cannot be done in InDesign.
In Acrobat Pro X, XI, or DC, choose File > Save as Other > Reader Extended PDF > Enable More Tools...
Thank you for this advice. Very helpful.