Demand curve for items where consumers typically only buy a single product This question began with rental housing (most people do not rent multiple apartments).  

The demand for apartment units in any given period is limited to the number of people searching for units (not all searchers must match with a unit in each period--assuming that most begin searching while their current lease is active).  Supply in the rental market is also fixed in the short term.

I expect that using the normal axes (x: Q, y: P) there would be a kink with a vertical line to the X-axis at the lesser of A) the units available B) the number of searchers.  At a certain price all of the searchers/units are matched. Is this correct?

Is there a different unit to use on the X-axis other than quantity (count) when consumer typically only purchase 1x of an item?
In the simplest case, to draw a two dimensional diagram, you need to assume that apartments are homogeneous.Suppose we go along with this. Then the market demand schedule can be written

$$Q^d(p) = \sum_{j=1}^N I\{w_j\geq p\}$$

where $N$ is the number of persons looking for an apartment, $p$ is the price and $w_j$ is searcher's $j$ "willingness to pay". $I\{\}$ is the indicator function, taking the value $1$ if the condition in the curly brackets is satisfied, and the value $0$ otherwise.

So as long as the price is above its willingness to pay, a searcher won't "demand" an apartment (he won't be willing to rent at that price, that is).

Strictly speaking, this will make the demand schedule a collection of distinct points, rather than a line, since indeed, the quantity axis is in discrete values. But since the price variable is essentially continuous, we can imagine that there exists a price where the number of units demanded equals the number of units supplied. 

Also, be careful with the assumption of a vertical supply curve. Ιt does not just imply that the available units are fixed in the short run, it also implies that apartment owners are willing to rent at any price.

This does not appear to be the empirical case. If prices fall, owners are seen to "withdraw" their apartment from the market, waiting perhaps for better times. This means that while we can imagine that eventually, we will have a vertical segment in the supply curve, for lower price we will observe a rising supply curve as usual.

For the rising part we will have analogously, for $M$ owners

$$Q^s(p) = \sum_{j=1}^M I\{s_j\leq p\}$$

where $s_j$ is the minimum acceptable rent by the owner. So an owner too is really "active" in the market and contributes to the supply only when the prevailing price is above that threshold (assumed possibly different for each owner).

A normal situation will be