The biggest consumer of art in the industry is Magic the Gathering by far... The article below suggests submitting by email, which is probably the only way to get in. They have many, many artists working on their stuff.
Here is the link: How to become a Magic the Gathering artist
The next biggest consumer is probably Fantasy Flight Games in Minnesota. I believe they are now the largest game publisher in the US and recently became part of Asmodee (from Europe). There is an email in the link for them as well.
If you live near Indianapolis I would suggest going to Gen-Con one year, it is the biggest Board and Role-Playing game convention in the country (60,000+ attendees) and they have a large artist ally and you can connect with many of the publishers.
I attended a seminar talking about how to connect up with game company art directors. Which is email or conventions and have a good business card (it should have a memorable piece on it and have good contact info.) The small publishers need to connect with artists too and most games involve many artists working together. One thing to note about small publishers though is it may be a while before they are publishing a game where your kind of art would fit.
Anyway, I have looked into game art and many of the artists are independent just like in children's book illustration.
I hope that helps a little bit.