Nothing quite that dramatic. Oil in Mexico is nationalized and the only company that can actually extract oil within Mexican territory is PEMEX, which is a government owned company. Recent reforms allow private companies to perform many tasks on behalf of PEMEX, including extraction, but it still maintains a national oil monopoly under the law (simplified: your company can go to Mexico and drill for oil on behalf of PEMEX and get paid by them to do so, but the oil still belongs to PEMEX until it sells it to you).

That said, a lot of the revenue for the federal government comes from oil exports. This further means that very little of PEMEX revenue gets re-invested into PEMEX, since it mostly goes to pay for what would otherwise be a huge federal deficit every year. It is also widely believed that there are strong political pressures from the U.S. not to reduce oil exports. So, in general, it's not easy for Mexico to prohibit oil exports without creating a huge crisis and/or majorly affecting trade relations.

The above becomes even more complicated when you add things like corruption, shared/disputed territorial water reserves (the Gulf of Mexico has large areas which fall under the territorial waters of the U.S., Mexico and Cuba, but access to oil deposits in those areas is not necessarily equal...) or contraband of "stolen" oil by drug cartels. But at a large scale, the main issue is: the federal government depends on oil exports and trade with the U.S. which might be partially conditioned on oil exports.

(Disclaimer: I am going to add to my posts that all this comes from the perspective of someone who was born and lived most of his life in Mexico and follows what I believe to be reasonable news sources. I am not a subject expert on: organized crime, Mexican law, Mexican trade relations, etc... you get the idea.)

How would preventing oil export harm the cartels?