Hello all, new to the forum and would appreciate any insight.

I thought I had a leaky pipe below my kitchen sink but we have confirmed there are no leaks.  The cabinet floor is quite damp.

Beneath the cabinet there is a toekick vent that is not attached to a duct.  When I removed the vent I can see a duct coming up through the subfloor.  The space beneath the cabinet is acting as the plenum for the AC.  I imagine the hot water pipe in that space sweats and the moisture is dampening the board.  Has anyone encountered this before?

Get a tin-knocker to make you a duct to go from floor vent to toekick outlet.  Given the location, a flex vent might be the only answer. 

Good job identifying the issue. Many would have assumed a leaking pipe.

It’s probably possible to fix this yourself and MAY be possible with standard HVAC fittings.  Unfortunately, I think you will have to remove a section of the floor of the cabinet in order to make the repair. If well planned, one can remove the section, retro fit the duct and replace the removed piece.

I imagine the hot water pipe in that space sweats and the moisture is dampening the board.

Sweating is water condensating on a surface that is below the dew point (temperature) of the ambient air.

Likely the hot water pipe isn't your problem as it'll likely be warmer than the air (= no condensation on it), with a cold water pipe this will be much different.

I suspect your AC is pulling outside air in, cooling it down, then releasing at the location you have the problem.

And with the cooling there is your problem: it reduces the temperature, through this the relative humidity if the air increases (up to it being saturated or, when cooling further, even directly condensing inside the AC - that's why they need a drain when in cooling mode), with that the dew point goes nearer (or to) the outlet air temperature of your AC.

Cool saturated air down even a little (or pipe it into an area with surfaces at a lower temperature) and you'll have instant condensation.

So let's say you have a hot damp day outside and you AC on to have it comfy, your AC most likely will feed your home with water vapor saturated air, which is no problem as long as the inside is warmer. But after the AC ran long enough (to cool the box it's venting into to the target temperature) you some °(C or F dosn't matter) more inside would be nicer and turn up the thermostat: the air from the AC will come in warmer but still saturated, and sees a colder surface...