While it might be possible to use the Click PLC power supply, I never suggest using the same power supply to power both Electronics and Coils. Coils cause too much interference and if not properly snubbed will eventually kill the electronics. Power supplies are so inexpensive any more it just makes sound practice to use two.
Which coils? Not interposing relay's coils? The Click high current block is direct connected to the Click so that should be protected by Click, right?
The two remote relays are in other equipment and cannot be changed to 24 VDC. The six solenoid valves likewise cannot be changed to DC.
(This is a retrofit to provide time delay due to a change in the process plumbing.)
The distance should be fine assuming a proper wire gauge is used.
The wire gauge is 12AWG for all conductors outside of the panel.
Battery Backup, this is easily fixed by changing the retentive ranges in the PLC.
retentive ranges I have no clue what that means. Does the Click have that option?
Proper Sizing and Protection - Yes. All coils should always be snubbed. More so when it is DC powered.
Where can I get the right snubber? The solenoid is marked "5A 120VAC". The remote relays are not marked but are also 120VAC. It is possible to install the snubbers on both the relays and the solenoids close to the coil.