Given this information...
... how can I find the thickness for a "NASA/Langley LS(1)-0417MOD general aviation airfoil" at 90% and 99% chord?
You have already done half the work. With the precise name it was easy to find the coordinates here and plot them. Yes, the coordinates look fine (no steps, no distortions)
Now either measure the thickness directly from the plot or subtract the lower contour point at 90% and 99% from its corresponding value on the upper side. You will need to interpolate between the two adjacent points, though.
Since you seem to need the thickness to calculate the lift curve gradient, you might just rely on the NASA technical paper which contains measurements and airfoil data on just this one airfoil. Be sure to correct for the proper Reynolds number and check whether your application uses the same thick trailing edge that is used in the coordinate file.