In short, there is no way to seal that concrete atop rock. Concrete is porous, and has low tensile strength (though high compressive strength), so that water can burst though it.
Plug one spot, and it will leak elsewhere. Paste over the whole foundation, and water pressure might even spall off some of the surface. If the leak were along a foundation wall, it would be a feasible DIY project to dig out the wall and place a vapor barrier outside the foundation. However, it is not a simple DIY project to break up the floor and install the vapor barrier underneath.
So what can you do?
Best is to prevent water infiltrating from outside.
- Landscape so that water runs away from the foundation.
- Direct rain gutter downspouts away from the house, and install an extension, such as one of these.
- Avoid watering plants near the foundation.
- Install French drains around the perimeter.
- If the water table is high, provide a way for it to flow to lower ground.
However, for the immediate issue, you might try using a portable pump that can be used on the surface of the floor, such as this.
Without a sump cut into the concrete floor, though, a pump cannot pull up all the water. The one cited claims it can draw water within 1/8" (~3 mm) of a surface, which still leaves standing water. Further, the water depth would not be sufficient to activate a float switch, so you would need to monitor the pump, turning it off when the water level drops.
Even with a sump, water would still flow over the floor until it reaches that lowest spot, leaving the basement damp, so best is to stop it outside the house.
Items cited are for illustration, not recommendation.