What is this part of a bicycle trailer called, and where can I find a replacement?
Bicycle Trailer hinge part.

Yes the original is moulded in Plastic but quite easy to replace with a piece of metal.

take a piece of aluminum tube that fits either inside or outside your present tube. Drill a hole through the tube close to the end for the bolt to pass through. Put a bolt through the hole so the hole will not be distorted, then tap the end of the tube down so that it forms the shape that will allow it to hinge up and down without fouling on the  square tube frame of the trailer.
you will have to cut out the rivet that holds the plastic piece in place and replace it with a bolt, use Loktite to stop nut getting loose.

An alternative, use a piece of flat steel about 1.2mm thick and 20mm wide i guess 150-200 mm long. cut out the rivet that holds the plastic moulded part in place. fold the steel into a u shape so it fits around the outside of the damaged plastic part, mark where to drill holes so that the steel strap now holds the damaaged plastic into place and a new bolt goes through the steel and through where the old rivet was.
enjoy cycling again. P