If the proposed project will require working with a contracting firm, should contract be procured during the three-year grant performance period or can an existing contract (competitively procured prior to the awarded grant) be used for the work?

This scenario would still be considered a noncompetitive procurement from EPA’s perspective, because you'd be using an existing contractor at the time of award. EPA’s recommendation would be to hold off on the actual procurement, using a competitive procurement approach like a Request for Proposals (RFP) or sealed bid, until after the grant award is issued. 

In the event that there is a contract in place, you would have to provide a rationale for use of noncompetitive procurement as allowable under 2 CFR 200.320 (c), which would be subject to EPA approval.  Or you would have to prove that there is something within your state or Tribal law or policies that explicitly allows for this existing contractor.