Will low oil pressure or failing oil pump cause fuel pump to turn off or quit sending fuel to rail

Have you performed an oil pressure test using a manual gauge to see what's really going on in the engine? That's the first step. Oil pressure does not feed fuel pump operation. That said, have you performed fuel pressure testing? Oil pressure doesn't show until the engine is running - so it's not about "key on" - it's about engine running. Use this link to find some shops near you that can help with your concern: www.repairpal.com/repair-shops