What does it take to get an ESRB rating for my game?
The primary point of entry appears to be the ESRB's website. The process appears to be that you need to make a physical letter on corporate letterhead with all sorts of information proving you are a real business. You then need to send them the letter and wait 5 days at which point they will send you a user/password for their website, at which point you can request a rating, which will take time and some amount of money.
For anyone looking at this in 2020, the International Age Rating Coalition (IARC) website gives you the info you need. If you have a digital copy game, and submit your game to a place like the Google Play Store or Nintendo eShop, you will fill out the IARC questionnaire. A game rating will immediately be assigned 100% free of charge for all regions where your content is available. It does not cost money. I repeat, you do not pay the IARC, ESRB, PEGI, etc, for anything. The only thing that costs money is development licenses which are issued by the digital storefronts, and not by the IARC, ESRB, PEGI, etc.