Do not use a mailing address or post office box (P.O. Box) that is different from your home address. There is a separate section on the form for your mailing address.
If you have no permanent address, you can use an address/location that will accept mail for you (such as a friend's or relative's home, church, or food pantry), and register from that address. You need to give an address so that your elections office knows where to send the confirmation telling you the location of your polling place. Most jurisdictions also have addresses for people who are homeless to use.