It sounds like you have a bad alternator, control module or pump unit. To rule out relays and electrical wiring issues, you need to take some voltage readings. Readings at the main ABS connector should read as below:
-Pin 1: should read ~14v with car running. Will not have voltage if main relay is bad.
-Pin 10, 20, and 34: Ground.
-Pin 15: 14v with car running. This is the charge indicator from the alternator. If this doesn’t check out, you have a charging issue which could be bad wiring or a bad alternator.
-Pin 25: 14v when you activate brake switch; no voltage with brake switch open.
-Pin 29: .65 to .75 volts; this is the green red wire on the negative side of the warning light.
If the issue is not charging, relay, or wiring related then your problem is with the pump unit or control module. You’ll need to swap with known good units to isolate the problem.
While you are checking electrical connections, you might as will check your wheel sensors. Each circuit should measure about 1000 ohms (rears 1100ish); FL-Pin 4 and 6, FR-Pin 11 and 21, RR-Pin 24 and 26, LR-Pin 8 and 9. A bad wheel sensor is obvious as the light initially goes out, but comes on after moving 50 ft or so.