My question is this: if, for example, I have multiple radars across an area, tracking a particular airborne object, can I use the returns from the radars to determine the geographical location of an object in three-dimensional space (its x-y-z coordinates)?
Of course. That is, after all, the whole purpose of radar: to get the position of an object and plot it on a screen. You don't even need multiple radars, one radar is enough.
If you're just talking about primary radar (i.e. bounce a signal off of the skin of the target), then you're going to have some trouble with measuring the altitude. The distances involved mean that a large change in altitude would create a very small change in the beam angle, so altitude measurements are going to come with huge error bars. In fact, civilian radars almost never bother with vertical scanning at all, because such imprecise measurements aren't considered to be worth the extra complexity. Naturally, secondary radar (i.e. transponder interrogation) doesn't have this issue, since the transponder simply tells the radar what its altitude is.