I think the only W116 that ever had a fuel pump relay was the 1980 model due to use of a O2 sensor, so only the M117 V8 got the fuel pump relay no M110 6 cylinder.
I tried replacing my fuel pump relay on my '78 450SEL when I overhauled my fuel system only to find out I didn't even have one.
Well my problem is the fuel pump is getting power, it just doesn't pump fuel. I replaced with pump with a new one, and the same thing happens. The pump runs fine when it is out of the car and connected to power. In the car, they do nothing.
Sounds like the strainer screen filter at the bottom of the fuel tank is plugged up. Has it ever been replaced as well as the main fuel filter?
Replace it along with the main fuel feed hose going into the pump.
How much voltage are you getting at the pump??? Should be about 12.5V with the engine at idle. Have a helper goose the engine while you're testing voltage at pump. Voltage should stay between 12-14V.
If the engine runs but seems to die due to lack of fuel, then usually suspect the fuel accumulator if you have CIS injection (a.k.a Bosch K-jetronic). The rubber diaphragm in the accumulator goes bad with age causing fuel pressure to drop off.
Stupid back yard mechanics. The PO put a new pump in without the check valve. burned the pump right out. I got a new pump and dizzy in. The original problem was never the fuel system. It was a small crack in the vacuum line to the air meter valve.