Does LSA require and electrical system?

No.

I have yet to see a handheld transponder for GA.

And you won't. The new system eliminates transponders.

There has to be a transmitter of some type installed to broadcast.

From: http://www.gisdevelopment.net/techno...s/ma04082c.htm

You still will not see it available as a handheld system.
It will have to be installed and certified in an aircraft with an
approved electrical system.