As of this morning, Thursday, June 30th, here is the situation. There is little, if any fuel available, not even through illegal black market sales, in most of Oaxaca and much of Chiapas. The refinery in Salina Cruz, Oaxaca is idle and all the fuel transport trucks are parked. Yesterday, refueling of Pemex stations in Chiapas via the refinery in the south of Veracruz was a failure as the trucks could not pass the highway roadblocks. Major routes have been blocked for quite some time and the situation is critical in supermarkets. Local markets always have food, locally grown produce and meats, fish, etc... available, however big commercial outlets are not stocked. Tapachula, Chiapas had only one Pemex still operating last night at 11pm but it likely is now closed.
Though you might be able to thread your way into this part of Mexico, you will at some point be faced with some major problems.
First, emergency services either do not have fuel and are conserving what little they have.
Second, if you do not speak or understand even passable Spanish, you are at a disadvantage to staying up to date with the local and state situations.
Yesterday, the Mexican Secretary of Government Osorio Chong, clearly stated the threat of using military action to clear the roadblocks if the ongoing dialogs are not providing solutions. This was quickly interpreted as a "sangre y fuego" situation (blood and fire) and the polemics of the events have now been ramped up, considerably.
The next two to three days are going to be crucial in the talks in Mexico City and the states of Oaxaca and Chiapas at the roadblocks. The situation is much more complex than is being reported, local information I get paints it as much more volatile and difficult situation. The military also need to have their plan and supply chains in place before going in or they will be stuck and isolated. Numerous deaths have already occurred and more use of deadly force is a definite possiblity.
I am not being an alarmist, it's that people outside of Mexico and people who are not tapped into Mexican media in Spanish are not getting the full picture.
Food basics are now being offered by the government in order to break the roadblocks but that can also further exacerbate the crisis.
Resupply of fuel to Pemex outlets and restocking stores will be a priority and will take a good 48-72 hours. Traffic, if roadblocks get opened, will be very heavy, mostly transport trucks. There has been no bus service to Oaxaca for days, now.
If you are planing on visiting these areas, give it at least 2 days for things to stabilize once the roadblocks are opened. Depending on how the roadblocks are opened and how much force is used, you might also be best advised to stay away for a week.