In theory, the elasticity quantifies in proportional terms the "reaction" of the dependent variable to a change of the independent variable, where the two are related through a functional relationship. So the concept is universal and mathematical -it applies to any univariate function, and not only in the field of Economics (when we have a multivariate function, then partial elasticities can be defined and used, but they face some constraints). So there is nothing "intrinsically aggregate" about the concept. In its theoretical conceptualization.
But in order to estimate/calculate the elasticity of any function, we must have the function in the first place. So the question boils down to "Can we obtain an individual demand function?"
It would appear that the matter is only one of data availability. If we somehow could have data on many transactions of the same individual related to the same good, then we could use these data points (price, quantity) =$\{(p_1,q_1),...,(p_n,q_n)\}$, in order to approximate its demand function... and we would probably be mistaken.
And this is because these points do not necessarily represent movements along the individual demand schedule. They are equilibrium points, points reflecting not the preferences of the individual alone, but their "crossing" with the supplier's behavior. Only if we could be reasonably certain that for a given set of transaction data, the demand schedule have not shifted (due to say, income effects), we could then use these data to estimate the demand function (by the way at market level, this is the classic endogeneity problem: using market outcomes to estimate a relation we estimate neither market demand, nor market supply, but rather the locus of market equilibrium points).
So while realized transactions possess the strong element of "truth" (the individual did actually bought such quantity at such price), they do not necessarily reveal to us his demand function.
In your case, if you can have other data on the buyer, especially in order to assess any possible "income effects" (for the case you are examining, that would be turnover and/or profit data for a company), you could combine them with its purchases from you to arrive at an elasticity estimation (there are also possible issues with substitution effects, but for them you will have to have data on what your customer purchases from your competitors). Is it a listed company? If it is, I guess you can find quarterly Financial Statements over the web (meaning that you will have to transform the data on purchases to quarterly too).
This could take the form of a multiple regression model
$$E(\ln q_i) = \beta_0 + \beta_1\ln s_i + \beta_2\ln p_i, i=1,...,n$$
where $q_i=$ quantity purchased in period $i$ (month or quarter), $s_i$ would be the turnover of the customer (assumption = if the customer sells more to its customers, he purchases more from me), and $p_i=$ average price of $q_i$. One could also include Customer's profit data as an additional regressor, in order to control for their possible effect on purchases from us. And of course, any other regressor that you think can have an effect on the purchasing decisions of the customer.
The use of natural logarithms, means that
$$\hat \beta_2 = \frac {\partial E(\ln Q_i)}{\partial \ln P_i} \approx \eta_{q,p}$$
i.e. it is the estimate of the price elasticity of demand. The above model assumes a demand function that is characterized by a constant price elasticity. You can make it more complex or even non-linear, if you suspect that the elasticity of demand in your case may vary.
