You don't need a diagnostic machine with this car if it's US spec...do the stomp test. Turn the key to position 2, don't start the car, rapidly press the gas pedal all the way to the floor five times within five seconds. It's hard to get the timing just right, but keep trying...it'll work.
I know it sounds stupid, but then the CEL will start blinking codes at you. It's not gospel truth, but it can really help out at times. I'll bet you see a 1221 lambda sensor code from the O2 sensor. You can splice in a cheap Ford Mustang one to replace it.
Google 'BMW stomp test' and something like 112,000 pages will give you the codes & more detail. This computer was installed in a ton of BMWs, they all work the same way until OBDII in 1996.