One side has 2 DC terminals for the trigger and the other has 2 terminals for the AC output. But the way the wiring is at my house is I have two 115v hot wires and a ground coming from the breaker. and the pump expects two hot wires and the ground.
Can I use this relay or do I need to find another? If so, can someone show me how to wire it?
For safety reasons (and electric code reasons), you need to switch both hot wires in a 230V circuit. You can do this by buying a dual solid state relay which isn’t cheap, or you can use 2 identical single pole solid state relays where each one switches a leg of the incoming power. Make sure the input sides of the relay are being controlled by the same I/O pin and circuitry so they are acting at the same time.
From a functional standpoint, you can make the pump run with just 1 relay since it will break the current flowing to the motor. The problem is, in the off state even though current can’t flow through the motor, it will still be energized to at least 115V due to the other leg not being disconnected. You can probably guess the safety concerns about having equipment that’s energized but appears to be shut off. This is why the electrical code and common sense dictates switching both hot wires to your load.
Crydom and Honeywell both make dual solid state relays. Some can be had for under $50 on Amazon but I would expect normal prices for brand new parts to be over $100 each. I would probably just use 2 single pole relays since they are more readily available and less costly. Try and use zero-cross relays to save wear and tear on your pump.
I should also note that you can sometimes get voltage leakage across a SSR. Often times a SSR will be used to control a contactor which provides a full mechanical disconnect when in the off state.
Voltage wise you’ll be OK so long as you are connecting the inputs in parallel. You will have to find a datasheet for the SSR to see what the input current requirements are and match that to what Pi can deliver. If the Pi’s current sourcing capability is lower than what 2 parallel inputs will require, you’ll have to use a transistor between the Pi and the SSRs to handle the drive current.