I can show my ignorance here by asking you to tell me what NORM is.

XRF is a bulk technique with a large homogenised sample. EDX is micro-analytical method usually analysing an area of approximately 1um.

NORM is an acronym for Naturally Occurring Radioactive Material and typically includes U and Th (and their radioactive daughter products, or progeny) and potassium, specifically 40K, the naturally occurring radionuclide of K. 

Uranium, Th and K can be determined by XRF in rocks and ores at concentrations ranging from ug/g (particularly for Th & U) to wt% levels. As noted previously in an earlier comment EDX (Energy Dispersive X-Ray) analysis relates to elemental analysis on a micro-scale using a high energy focused beam of electrons to generate X-rays in a sample - the energies of the X-rays are characteristic of the elements in the sample and their intensities are proportional to the concentration. EDX is typically only applied to the determination of major and minor elements and not at concentrations less than several hundred ug/g (i.e., trace levels).

In summary EDX could be used to measure the NORM elements U, Th and K if abundant and one wished to know their distribution on a micrometer scale. One could employ XRF to determine the elemental abundance in bulk samples at the trace elemental level and/or higher concentrations. 

   Neither XRF,  nor EDX, could be used to measure the 'common' NORM radionuclides 226Ra, 228Ra, 210Pb, 228Th, etc (products of the 238U and 232Th decay series). Other techniques, such as gamma-ray or alpha-spectrometry would need to be employed. 

   Hope this helps.

After posting my answer yesterday I did add a comment /question as to whether the original question about 'NORM' in fact was referring to CIPW NORM calculations (based on XRF or EDX major elements data), i.e., normative mineralogy calculations! Unfortunately that post doesn't appear to have made it.  

So, if the original question related to CIPW NORM calculations using either XRF or EDX generated elemental data then again, one would wish to use bulk major elemental analyses provided by XRF vs. the EDX measurements. As the XRF analyses will be 'sampling' a much larger sample than EDX the CIPW NORM calculations will be more representative, and hence useful, when based on XRF data. Again...hope this helps.

The confusion emphasises the importance of stating questions as unambiguously as possible.

To add to what @Duke says above:

In XRF bulk major element is done using fused sample pellets (for minerals or rocks its Std practice). However you can also get minor and trace elemental concentrations using powdered pellets made with the help of binders or palletizing aids. So the heterogeneous composition of the sample can be checked by SEM-EDX while XRF will treat the sample as a homogeneous bulk....its almost what @Duke already stated above :)