CO2 dissolves very readily into water, but it also "outgasses" very readily. So, an open container of water will always be dissolving CO2 from the atmosphere and losing it back to the atmosphere at the same rate. When we diffuse CO2 into our aquariums most of that CO2 soon escapes back to the atmosphere, so to maintain a high concentration of CO2 in the water we have to continuously diffuse more CO2 into the water.
If you relied on the high CO2 content of the ground water for CO2 in your aquarium you would have to have a continuous flow of ground water into the tank, with an overflow to avoid overfilling the tank. As a guess I would expect to need around 100 gallons per hour for that flow for a tank that size, in order to maintain something near 20-30 ppm of CO2. I'm basing this on the fact that when we use a DIY external reactor in our filter return line we are running about that much flow, and that flow may have about the amount of CO2 dissolved in it as your ground water.
Brief answer - it would work, but it would waste enormous amounts of water.
EDIT: After a bit more thinking, for a low light tank, a little bit of CO2 could be enough. So, if you were trying to get something closer to 10 ppm than our usual 30 ppm, you wouldn't need as much water flow. And, that suggests an interesting test that could be done. Use a pH probe to measure the tank water pH. Then collect data on tank pH vs. groundwater flow through the tank. Finally, use a drop checker to get a one point "calibration" of tank pH vs. ppm of CO2 in the water. With that you could determine what water flow rate would maintain the tank at 10 ppm, or 5 ppm, or whatever you wanted to shoot for.
But, suppose the concentration of CO2 in the ground water is seasonal? Oh well........