What steps should I take to figure out which alarm model I have and how to go about installing a missing sensor?
It's surprising not to have a brand name visible, either on keypads or the main system. I will say it is possible that your system is very outdated, and so the alarm company basically doesn't want to touch it (rightfully so), they just want to put in something modern which they know they can support if it breaks.

You should have a control system somewhere. Often these are a metal box that opens and has a circuit board inside. There should be a model number or name on that, and if not, posting a picture would help.



If there is a circuit board like this, it is possible the model/name is printed on the reverse side, as they actually sell the box separately from the circuit board -- the installer mounts it at install time. You'd have to remove the circuit board in order to check this (cut the power, and be careful not to damage the board -- including being careful of static).



Aside from that, there are three general ways to go about fixing the problem.


Install the missing door sensor


Installing the sensor may or may not be easy. If you have another door sensor, you can verify that you're getting the same thing (usually just a NC magnetic contact sensor, like these):

  


Short out the door sensor wire


You may be able to short out the door sensor. There is a very small chance that this won't work, if it requires a resistor to terminate, but that is not common on door circuits. Of course, this door won't be alarmed if you do that. It also means you need to find that wire, somewhere. You could do it at the panel, as well.


Remove the door sensor circuit


This means re-programming the system to not use this circuit. It definitely requires finding the model, and then figuring out how to program the system. Sometimes this can be done through (at best) cryptic button sequences, other times it needs PC software and a (sometimes special) cable.