It sits on the passenger side engine compartment toward the rear of the vehicle. The white tube is very brittle and has broken off. Obviously a vacuum tube, but I have know idea what it is for.

 

That is the recirculation vacuum motor. It closes the outside air door on a/c max setting. The white vacuum line can be repaired with a short rubber line

 

How much would this affect the performance of my A/C?? BTW, I'll have to run the rubber line through the firewall because the white one is brittle all the way. In fact, I don't even know if the white line is brittle on the inside as well. I'm hoping not. We'll see. Thanks!

 

As said above it will only affect the performance of my A/C in the MAX position. In other words all the cool air will go to the defrost vent in MAX but NORM will work right.
It will also be a vacuum leak for the engine in the OFF position.

So you can not use the OFF or MAX settings.

 

It does help performance by recirculating cooler cabin air,especially on hot humid days . Usually the white plastic line disintegrates where it is exposed. You need to add the line to a section that is not disintegrated.

 

That vacuum line doesn't go straight through the firewall inside the cab, it first runs towards the bottom right hand corner of your photograph into a little bundle of vacuum lines.
I replaced that same line just a couple weeks ago with a couple feet of rubber hose. The existing white line was in good shape inside the tape wrapped bundle of lines so all that was needed was to couple a new piece of hose to it.