The turn signals on my AT3 are not working. I took the switch apart and found that all three wires are no longer soldered on to that plastic switch plate.  My question is can this be fixed or do I need to get a new switch for the left handle bar?  

Usually can be fixed.

If you don't want to tackle it yourself, DEET does it.

+1 to Mark's reply.

If the bakelite style plate that holds the wires/terminals is broken, you may be able to find a damaged switch for parts - many models used a variation of that switch body (YHA205 cast into the underside), so it's not just an enduro one that you need to look for.

Some Kawasakis of the same era used a near identical switch to this Yamaha one, and they offered individual components as spares, although the wire colours will be different. As an example, here are the complete harnesses, still available from CMSNL.

WIRING HARNESS-L.H SW for Kawasaki - order at CMSNL

I have rebuilt a few of the Yamaha ones over the years, and bought new silver/grey sleeving from DEET. Obviously the bullet terminals need sacrificing when replacing the sleeving, but new ones are readily available.