Thanks for the pictures–that really helps!
So you have a 12V DC coil relay with 30A 250V contacts–a very standard part. If you are replacing this entire box, you can get a new relay, but you could also re-use this one.
Motors have starting capacitors which in your case looks like they are in that box. A lot of motors have centrifugal switches that turn off the starting caps when the motor is up to speed, but this motor might not have that. I see heavy wire going to the starter switch which makes me think the motor does not have the centrifugal switch.
Do you have to hold the green start button for a second or two to start the pump normally? If so, you will need a starting relay and a running relay to duplicate that way of working.
You are going to need all those starting capacitors that are in there now. It is unusual to have so many and to have some added in the bottom of the box like that.
The red stop button looks like it is only low current or low voltage and probably interrupts the voltage to the coil of the LEONE 30A running relay.
The 30A analog meter is not telling you the actual starting LRA. You need more sophisticated test equipment to get the real number, but if you have a 30A relay in service now, you need at least that big a relay in the future. Relays are a mechanical part that wear out and need to be replaced periodically, so they are always available.
That is actually quite a big project and very nice to see!