You still must pickup your permit in person no sooner than 48 hours in advance. For your safety and the well-being of black and grizzly bears in the park, you are required to camp in a designated campsite. If that scenario doesn't effectively demonstrate the inadvisability of Fido in the backcountry, note that Yellowstone has excellent coverage of the backcountry with ranger patrols and they will give you a ticket. The most common cause of death and serious injury in the Park is falling into thermal features. Hot potting in a thermal feature is illegal in Yellowstone National Park.
Thermal features are highly variable in temperature and are extremely fragile.
Backpacking Yellowstone National Park has over 1100 miles of trails, from varying levels of difficulty and length. Backpacking through Yellowstone allows an unparalleled opportunity to experience an amazing diversity of terrain, wildlife and plants. If you are unfamiliar with the area and planning a backcountry hike in Yellowstone, a good way to get an overview of possible hikes is to start with any of the various guide books available from book stores or your local library. The book covers over a 1000 miles of trails and includes information on how to hike through bear country safely. Written by a Yellowstone backcountry ranger and hiker with 25 years of experience, this guide is highly recommended by the backcountry office as the definitive work on the backcountry trail system.