score:4
You do need a permit, and per the last Space Availability Report, Bright Angel is booked immediately after Christmas, but there may still be availability (the report is from a few weeks ago) earlier in the month. If you cannot get a permit, it's possible to try to for a last-minute permit in person (show up early) or the waitlist; details are on the website you linked.
I would apply immediately by fax (you can use an online service to send the fax from your computer if you don't have easy access to a fax machine) and they'll respond by email.
I don't know your winter hiking experience, but ensure you are prepared for the conditions, which may include considerable amounts of ice and snow on the trail, winter storms, and water not being available during the off-season in some areas. Make sure you have the necessary equipment.
This page has some information on transportation that may be helpful, but we don't know where you're coming from. That page recommends outfits offering both scheduled and on-demand transportation services and taxis. Without a car, careful planning will be required to avoid becoming stranded, especially in winter.