Yes you're correct the 5240 and 5640 are slightly different. We still need to check the starter solenoid first which will likely be back in the engine bay somewhere. On a 5240 you open the engine cover and it's on the left side. Here is the parts manual for your machine:
https://midtnequipment.com/content/gehl-parts-manuals/Skid-Loader/SL4640-SL4640E-SL4840-SL4840E-SL5640-SL5640E-SL6640-SL6640E-Skid-Loader-Parts-Manual-917000J.pdf
#23 on page 191 is what we need to test and is pictured below. There should be two of these. One for the glow plugs and one for the starter and you may hear a click from the glow plug solenoid when you turn the key on. The starter solenoid should be right next to it. You can cross the terminals here to see if the starter turns over or swap the glow plug solenoid for the starter solenoid and see if the starter works.
Edited by tlarson4430 11/16/2020 20:38
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 solenoid (full).jpg (45KB - 61 downloads)
solenoid2 (full).png (131KB - 116 downloads)
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2012 Gehl 5640 skid steer no crank
