Sure. The KDE is a nonparametric density function (PDF) on a finite domain. To get the CDF from the PDF, Numerically integrate the PDF. See the article "The area under s density curve: Nonparametric estimates" The article uses PROC UNIVARIATE, but you can use PROC KDE if you prefer.
If you use a very fine grid for the KDE, you should be able to closely approximate the CDF and therefore the probabilities for any interval. You can use the NGRID= option to specify how many grid points you want in the output.
If you require interpolation, see the article "Interpolation in SAS" I think if you use lots of grid points (maybe1001) you won't need interpolation. After all, the KDE is only an estimate.
As you say, for each test value in the data base, you can get an approximate probability by finding the first x value in the linear grid that is greater than or equal to the test value and looking up the CDF for that x.
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