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It is okay to have a 12 V supply on the output of LM2937; however, typically the external supply would be set higher than the nominal regulator voltage of 12 V plus accuracy so that when the 12 V supply is present it will supply the downstream circuit. Please note that since in the real world it is unlikely that both the output of LM2937-12 and the 12 V external supply will be exactly 12 V, whichever has the higher voltage will be powering the downstream circuits.
Since traditional linear regulators do not sink current, if the output of LM2937-12 is lower than the 12 V external supply, pass element of the LM2937 will be off until the output voltage is reduced back to where the LDO will once again regulate the output.
Your application seems similar to the one discussed in the following blog series:
Now i feel Confident to have 8-12V on the output of LM2937. There will never in this certain application for this board be voltages on the Input of LM2937 since it is not connected.
