Basically, I want to use an Arduino to control this relay and get rid of the push button. So that the Arduino would give a command like the button does.
I was thinking about using of those 5V relay module for Arduino, but I'm not sure how to to do the wiring. How can I solve it?
I quick search implies that what you have there is a "flasher relay" - power goes in (B) and comes out intermitently on the L pin, causing (eg turn signals) to blink.
If that's the case, it isn't controllable with an Arduino.
If you wanted to replace the turn-signal blinking relay with something Arduino-controlled (so that you could change the blink rate or something), I think you'll need to find a normal relay (4-terminals or more.)
Indeed. A flasher relay looks the best fit for example: https://www.jaycar.com.au/12v-2-pin-universal-led-relay-flasher-module/p/SY4016
So it looks like the "push button" is in series with the load if the diagram in the above link matches what the OP has. In that case, a 5 volt arduino driven relay module could be used to replace the push button.