Can some wet/dry vacs indeed continuously pump water?
There are two solutions:

Wet/dry "Pump" Vac

Look for a wet/dry vac that is labeled "pump" (the manufacturer will likely draw a nice hose with fresh water prominently on the box).

Such wet/dry vacs have a built-in water pump. One model on the market looks like this:



Universal Pump for Wet/Dry Vacuums

These are add-ons to your wet/dry vacs. Your vac must have a downspout for this option to work. One model on the market looks like this:



Neither of these options will allow continuous pumping.

You cannot continuously pump out with the first option because the wet vac will be faster (more powerful motor, 1 1/2" to 2" hose) than the water pump (5/8" garden hose, pump handling just water rather than an air/water mixture).

You will need to pull the hose out of the body of water when the tank is full. Force the power button down to override the shut-off feature for 15 seconds. The water pump will then continue pumping out (at 350 gallons per minute) until the tank is empty. Repeat.

You cannot continuously pump with the second option. You will have to attend to the two pumps. Fill the tank with the wet vac. Turn off. Empty the tank with the water pump. Turn off. Repeat.