Yeah it can cause issues with the fuel pump because the lower the batteries state of charge gets,the less voltage its gonna have to feed the pump.Another thing about a battery is,once its state of charge gets below a certain voltage level, the alternator will no longer be able to charge it.Here’s why: To generate power, you pass a coil of wire across a magnet. It’s basic physics #101. In an alternator, they turn things around. They rotate a magnet past a stationary coil of wire. The key here is that the magnet is an ELECTRO magnet. That’s right, you need a good battery to provide the power to make the electro magnet work in the first place. An alternator cannot make power if your battery is dead. This is why you cannot charge a dead battery with the car’s alternator. No 12 volts in=No 13.5 volts out.So once your battery got below this level,the alternator stopped charging it.You could also have a loose or corroded battery cable at the block/timing cover & that would keep the battery from taking a charge too.So check your neg battery cable at the block to make sure its tight & corrosion free.Another thing you need to check are the two fusible links that are inline with the alternator wiring harness(14ga &18ga)If one of these links burns in two,it'll create an open in the harness between the alternator & battery & that would prevent charging.The best way to locate a burnt link is by disconnecting the pos & neg battery cables,disconnect the wiring harnesses from the alternator, remove the wire(that has the fusible links)from the starter relay(on the fender apron) Use an ohmmeter to test the wire.Touch the (+)meter lead to the alternator end of the wiring harness & touch the (-)lead to the wire you just removed from the starter relay.The meter should read 5.0 ohms or less.If it reads above this,the fusible link is blown.Use the wiring diagram below to locate the wire these fusible links are connected to & test the links.Another thing that will cause the alternator to not charge is the battery light on the dash.If it doesn't illuminate when you turn the key on,the alternator won't charge.You should be OK with this one though because you stated the battery light came on.
From my experience with the Foxbody,the part that normally goes bad is the brush assembly.You can easily replace these brushes yourself for $5....All you have to do is remove the alternator,locate the voltage regulator(it should have a few really small bolts*torx bit maybe* holding it on.Its black & it'll have a really small hole in it that you have to insert a paper clip/small nail into before you remove the bolts.The paper clip keeps the brushes in place so you can remove the regulator.You'll use the same method when you reinstall the regulator.If the brushes are really worn down,thats most likely your no charge issue.The pictures shown below are the regulator & brush assembly, just in case you're not familiar with them.
I don't like Autozone brand alternators,so if the brush assembly is the only thing that's wrong with it,you can just replace them & save yourself a lot of money.What I do is take the alternator to Autozone & get it tested.If its bad,buy the brush assembly,take it out to your vehicle,remove the old brushes & replace them with the new ones,bring the alternator back inside & get them to retest it.If it passes,you're good to go.If not,the diode might be bad instead.Its the first picture shown below(one of the wiring harnesses plugs into the diode & the other harness plugs into the voltage regulator)The only things that normally go bad on an alternator are the brushes,diode or bearings.
BTW-dont forget to bring a paperclip,nut driver(to remove the nuts that attach the brush wires)& whatever size torx bit/Allen wrench fits the regulator bolts,so you can do the replacement in the parking lot.