I have a 89 firebird formula 5.0 liter with throttle body injection. My fuel pump does not come on without cranking the engine for a couple of minutes. I have a B&M shifter in the car.

Replace the fuel pump. After 20+ years it's tired.

fuel pump is going bad or the ignition switch is bad.

Replace the pump it should be easy sense it's on the engine.

if he has TBI the pump is NOT on the engine it is an electric pump inside the gas tank the pump produces 13PSI to replace the pump the rear axle has to be dropped and if the car runs fine then chances of it being the pump are slim the pump will only run if the engine is running, cranking, or for 3 seconds when the key is initially turned on if the pump runs when the car is cranking then im assuming ur issue is the car starts the fuel pressure drops off and dies? try jumping the fuel pump with 12 volts to see if it stays running if yes ignition switch is a good place to start if it does not stay running then do a volume test on the pump the pump may be bad if it dies while being jumped with 12V it may produce 13PSI but does not put out enough fuel which the volume test will tell you the amount of fuel being put out

^john is right. it's a pain to get to that fuel pump but you probably won't have to replace it again at least. i'd say its a bad pump tho. its a lot more common than a bad ignition

check out the relay and then try the fuel filter.. then the pump if it still persists

You should be able to see if the injectors are working while cranking. Take the air cleaner off and have a friend watch the injectors for fuel to squirt out. If no fuel squirts out then rent a node light set from Autozone and make sure the injectors are getting a pulse