If my RAM chips were dislodged, could that be causing my PC to not function?

The pc won't be able to boot-up or even get a display output on the monitor with dislodged RAM. Or worst, dislodged RAM might fried.

A couple months ago I got a PC for free and all that was wrong with it was a stick of RAM wasn't fully installed. 

Reinstalling them correctly may fix the issue. You would have to try and see.

It should. ...........................

Reseat them and try again. Simple.

Let's take a look at a couple of your statements:

A) "I was told..."  By whom?  A tech?  some guy you know?  Never listen to anyone who does not have a full background in both building and servicing computers.  A BSOD does not necessarily mean you must replace the MB or the Computer.

B) "Dislodged"  Were they not in their slots? RAM Chips, when properly installed, cannot easily be 'dislodged.'  If you see them improperly mounted, then this is improper installation, not 'dislodging' by some other means. Not trying to point fingers, so let's move on to the fix: Unplug the machine, Carefully remove the improperly installed RAM and then RESEAT.  Now, if they do not go into the slot, make sure you are using the correct RAM Chips, orienting them correctly so that the notches line up between the Chips and the Slot. Put one hand UNDER the MB if possible so that your pushing down does not break or crack the MB. When Properly seated, there are usually a set of plastic clips which will be 'pushed' into place which then both hold the Chips in place as well as offering ease of removal should the need ever arise.  

C) "Will this fix my issue?"  Unsure, dependent upon the BSOD.  The machine didn't crash, so it may actually power back up once the chips are replaced properly.  

I hope this all helps.  Please post more if you discover further problems. 

If they were dislodged, the system would not boot at all. If they were not securely mounted and the system was bumped while powered on, that COULD cause a blue screen.