Hi and welcome to the forum.
You've had a spot of bad luck. This is most unusual; the ABS on the 500 is normally trouble free, so there isn't a great deal of collective knowledge on the forum about this.
I'd start by interrogating the car with a dedicated Fiat-specific scan tool to see what fault codes are stored.
IIRC the ABS pump is supposed to cycle each time the car is switched on; possibly whatever control mechanism is used to manage this isn't turning off correctly. Stuck relay, perhaps?
New ABS parts are prohibitively expensive, but used ones from the breakers are reasonably priced (there's not much market for them as they fail so rarely); the difficulty you face is probably going to be diagnosis and reprogramming.
If it is a relay,
and you can identify it,
and you can by some means open the case of whatever module it's in,
and you can find an equivalent relay (hint: try Farnell or RS components; failing that, swap the relay with one from a scrapped module),
and you have decent soldering skills, then just maybe you could fix this quite cheaply without needing to reprogram anything.
If you haven't already got it, investing in a licenced copy of
MES and the appropriate cables is probably the next thing to do. It'll likely cost you less than a main dealer diagnostic check, and you're almost certain to need it if you're going to replace any of the electrical/electronic parts. I'd be reluctant to go anywhere near a main dealer; the repair cost could easily exceed the value of the car. Most non-fiat general purpose garages won't have the necessary diagnostics, but an ABS specialist should.
And please come back and let us know what happens; we've very little information about these modules and your experience could help others in the future.