When I moved into my house 19 years ago, we encountered a similar situation. No sump pit, it rained about 8 inches two months after I moved in, flooded basement. I was successfully able to dig a sump pit and install a pump under water, here is how I did it.
First of all, you are going to need to break the concrete to install the pit. I had about 3 inches of water above the slab when I did the work. I used an electric jackhammer attached to a ground fault interrup circuit. I wore rubber boots, even so probably a pretty dangerous way to cut the slab. Better off using a pneumatic hammer to cut the opening, you can rent one. I did not have the money at the time.
I chipped out an opening about 24 inches in diameter, with the intent to install a 20 inch diameter perforated bucket, 24 inches deep, to collect water, which would be pumped out to a low area in my yard about 150 feet from the house via a 2 inch diameter PVC drain line.
Once the concrete was removed, I dug the rest of the hole by hand, with a flat bladed shovel. Since there was gravel beneath the slab, it wanted to collapse, so I used wood staves about 3 feet long driven into the gravel beneath the slab to hold the gravel back. I borrowed this technique from commercial trench diggers, who still use it to this day on large projects (the sheeting is locally known as Wakefield sheeting). Simply temporary wood sheeting to hold back the soil while digging.
Once I had the hole dug to the proper depth (24 inches), I inserted the premade plastic tub made from a plastic garbage pail, with 3/8 inch diameter holes drilled into it. The sump pump went into the bucket, and I connected the sump to a dedicated outlet with a ground fault interrupt. Fortunately the outlet was already there, I don't think I would have wanted to wire when the floor was still wet, which it was. I set the float level to turn the pump on when the water was 6 inches below the slab, and to turn the pump off when the water dropped to 12 inches below the slab. This keeps the pump wet at all times, which is necessary for a submersible pump.
I dug the trench for the 2 inch PVC line by hand, I only buried it about 16 inches, which is not below frost depth, but to dig below frost would have been a huge job. I initially used a backflow preventer on the pump, but after several years of periodic freezing of the line in the winter, I got rid of the backflow preventer, and the system works quite well. It has been working fine for about 18 years now. I use an industrial grade, 1/2 HP, 100 volt Barnes sewer pump with 3/4 inch solids handling capability. Due to the variations in groundwater in my yard, the pump typically only works three or four days a year, but it is quite effective in keeping the basement dry.
Now as for the comments about connecting the pump to an exterior drain line, this is a great idea, assuming your house has an exterior drain line. Mine does not, and it would be quite costly to install one. Because my slab was place on at least six inches of coarse gravel, water beneath the slab flows through the gravel to the sump, where it is ejected. If your slab is not built on gravel, this will not work, and you will need to use a different plan.
Based on experience, it would of course be easier to wait until the water recedes to dig the hole and install the pump, but it CAN be done in the wet. Be safe, if you use electric tools make sure they are double insulated and connected to a GFI outlet, even then I recommend using pneumatic tools with a compresser, much safer. Good luck.