Air Carrier First Officer reported flight crew missing required Hazmat 'Battery powered wheelchair form' in order to pushback from gate. After conferring with Dispatch and Flight Ops Duty Officer the correct procedures were followed and a completed form provided to flight crew.

Narrative

We had an issue with a wheel chair battery trying to be stowed in the cargo hold. The Captain arrived at the plane first. When I arrived the gate agent told me to tell the CA (Captain) that the wheel chair issue was taken care of". When I arrived in the cockpit and told the CA. He told me what happened that a passenger asked if she could stow her foldable wheel chair in the cabin possibly in one of the closets and that it had a Lithium phosphorus battery; and he told her no it would have to go in the cargo bin. We didn't think anything else of it until we were ready to push and the Captain asked the Ramp if about it. That started a discussion with him about the battery. We realized we needed a form but the Ramp didn't have it and they needed to fill it out as well. They had put the wheel chair in the cargo bin with the battery still in it. No one was real sure what kind of battery it was and what needed to happen.The Captain made a call to Dispatch and the Chief Pilot and eventually the hazmat hotline to try to figure out what to do. It was about this time the Gate Agent came to the window and handed us a wheel chair form. The Ramper said he never saw it and he needed to sign it. It became clear to both of us in the cockpit that the Gate Agent quickly filled it out just to get us something without following the proper procedure EG she pencil whipped it. Eventually they pulled the battery from the cargo hold and had the passenger carry it with her in the cabin and got the form signed by the Ramp and we were satisfied that we done everything properly. during cruise we read into it a little more and we think that we ended up doing everything correctly except that we don't think the passenger ever got a copy of the form for the battery which we think that she may have needed. We were not sure what the proper procedure is that needs to happen from when a wheel chair with a battery arrives at the gate to when it gets to the cargo hold. It seemed clear that the Gate Agent pencil whipped a form to try to get the plane off the gate. If the Captain didn't ask about it the battery may have gone improperly in the cargo hold."