Is it legal for a 501c3 non-profit group to enter into a cooperative legal agreement with a state agency?

You will have to be more clear as to what type of non profit group, the 501C3 is normally considered a church,
And what type of contract ( legal agreement)

I know of churches that have agreements to provide half way house services, that receive state money for homeless shelters, provide training

A non profit group is a corporation, so they can legally enter into contracts.

The issues come of course that once they start receiving state or federal money they have a lot more rules and restrictions put upon them

A 501(c)(3) is a non-profit and covers a wide variety of organizations. My best guess would be that the non-profit is doing the same type of work as the state agency, such as children's services, etc. This is really a corporate law question, but as someone who worked in non-profit for several years, I don't see why not.