“We want to play any constructive role we can play with federal resources in getting to the bottom of it and preventing it from happening in the future,” Psaki said.” [from The Guardian]
It means that the US government is willing to do anything they are capable of in order to find out what caused the building to collapse.
so “play with” means “make use of”
In this context, to play a role means the function someone performs within a group. Or you could look at it like the job they do.
For example in football (soccer) the goal keeper plays the role of protecting the goal. Other players perform a defensive or offensive role. The referee plays the role of making sure the players play by the rules.
to play any role we can [play] using federal resources
(the second word play is not needed)
not
play with federal resources
The city and state are primarily responsible for finding out what happened. The federal government is offering help in any way that might be needed.