Yes, they do. They also have Buddhist and Wicca chaplains, among others.
Chaplain : Chaplain & Support : Careers & Jobs : Navy.com
The question is, what about the chaplains who represent religions that view homosexuality as a sin? Will they be allowed to preach, teach, and counsel that view?
Wicca chaplains? For real?

Anyway...
Given that preaching against homosexuality could be seen as a form of discrimination, I think yes, that sort of speech would be done away with, and those chaplains not willing to change would be dismissed. Personally, this seems no different to me than not tolerating discrimination based on sexuality in the federal workplace.
I know that Chaplains are a necessity when the service members are overseas and fighting, but this is what happens when you tinker with Church and State. The Military is a function of the State, and the State cannot tolerate discrimination.