Hy aspirant, 

Esic quota is being applied for admission to neet 2019 mbbs and bds course already. You can refer this answer regarding admission through this quota eligibility- 

https://www.careers360.com/question-how-to-apply-for-esic-quota-in-neet-2019

So, the counseling for this quota depends on which state you belong as most of the state has its counseling going on but has its 1st round done and 2nd round going on than mop up round will be left. 

Depending on your marks and cut off of your state According to the no of total seats compard to no of reserved seat under esic quota,  you can apply for this during your counseling and if you fullfill the criteria under esic quota, you can get your seats in mbbs/bds. 

Apply for it first, as almost all rounds are getting over in all the states. 

If you make it this year, you can try for other popular medical courses like bams,bums, bvsc, etc or can try out next year again. 

Good luck