We can ignore specifics and more generic escape from a large city and massive problems of traffic and disabled power caused lack of traffic control and fuel availability.
What vehicle ?What vehicle related tools etc?,normal stuff please.
What supplies for the bugout or beyond?
depending on if you have an actual BOL
What preps for vehicle and ,lets say four in the family?
I know bugout has been done before but I think not with the logistics of mass evacuation and traffic problems as a part of the plan.
OK. Here are my thoughts based on my own preps.
However I have never had to actually bug out. I have only done practice runs. I relied heavily on the advice Jack has given but modified it for my own use.
Vehicle is a later model half-ton (150/1500 class), crew-cab pickup truck. It is 4X4 but
does not have an off-road suspension. I wanted a suspension which provided better handling on road. The transmission is also the middle option available (mix of decent gas mileage and tow capability). I have added a good Class III/IV hitch, trailer brake controller, and trailer plug adapter for easy and safe towing.
The bed is covered with three-fold
hard top (specifically the original extang solid fold). When coupled with a tailgate lock like the Pop & Lock this creates a large 'trunk' for carrying items. But it can easily be opened for carrying bigger things. Plus it is an excellent platform for a rifle. Just the right height for shooting over while using truck body for cover.



But this is not water tight. So I also carry an SKB iSeries waterproof case for the main gear. This thing is so well sealed that metal items placed in it don't rust. It also makes for a great camp bench/table.

In it I keep a lot of equipment. I have most of the items that jack talks about. But the key ones for keeping on the move are a small camp axe, foldable shovel, tow straps, hand winch (from harbor freight but works great), bolt cutter for gates/locks, 12v air compressor, lights, fire extinguisher, and gooloo jump starter/emergency light/USB charger. This last gets moved into the cab and is charged as I drive. The gooloo is an impressive piece of kit. With 18 amp-hours it can keep all my USB chargeable items items running for two weeks. This includes cell phones, transceivers, APRS devices, GPS, etc. It is sort of a cross between Steven Harris' vehicle backup and the tech in his cell phone video.
http://www.gooloobox.com/products-detail.php?ProId=50
This combination of items will allow us to quickly clear items from the road (e.g. tow straps to pull/axe to cut) and get us out of being stuck (shovel and winch). With this I still have plenty of room for carrying other preps. One big one is I use a cross-bar near the back (i.e. bumper side) to 'trap' five ~5 gallon filled gas cans in place. This doubles the range of the vehicle to 800 miles.
One nice thing about such a truck is that it has some off-road capabilities should the roads be jammed. In a place like Chicago this would be invaluable as one could (slowly) drive along the train tracks and power easements.
Regarding stowing of equipment, each person keeps their BOB near them (under seats in rear, on floor between seats up front) just in case the vehicle has to be quickly abandoned for foot travel. Most of the rest is stored in back...unless we are towing our off-road RV. But that is another story. As is my hams/murs/gmrs/aprs comms and hacking into the integrated onboard wireless phone system by using bluestar:
http://costartech.com/pb/products/bluestar.html