Drying Out the Basement

I would try a dehumidifier. We have one in our basement that runs 24/7.

Dehumidifiers work great!

The best you can do right now is to keep pumping/wet vacuuming it out.
Then run dehumidifiers to dry out best you can.
Run them through out the house to get the moisture out.
Unfortunately, it sounds like you're going to need to hire someone sooner or later or your foundation is going to go and/or you are going to have some serious mold problems (which can have some serious health side affects - our neighbors house was condemned due to black killer mold - the kids were throwing up blood amongst other symptoms).
Hiring someone before there are other problems will be less expensive in the long run.

sounds just like our basement - we use the shop vac too! Then I use these two really big canvas drop cloths (the kind used when painting) to sort of mop the rest of it up, then throw the soaked drop clothes, one at a time into my washer and dryer. Drop cloths are pretty cheap, like $10 each for these big ones. I do plan on buying a dehumidifier for our basement this week also though. We have been getting so much rain!

My husband works in that industry. It's likely covered by your homeowmer's insurance. Continued damage after the fact might not be covered if you let it go.

Definitely a high-quality dehumidifier. And install a sump pump if you don't already have one.

I use to be in the same position. Unfortunately you are not going to like my answer. The more it rains the worse your basement condition is going to get, with every heavy rain the cracks are getting bigger in your basement. In a year or so you may always get water in your basement with even a small amount of rain. I have been there.

Dehumidifier does work but we were running two ALL the time. The basement would kind of be dry but after every rain it still became wet.

Sump Pump ONLY works if you have a good clear/clean drain tiles either on the inside or outside of the house hooked up to it. We found our drain tiles clogged with clay hence why the water took the past of lest resistance through the walls & floor and into the basement. Cement has air pockets/pores so id does not block out water, the water will seep in, create cracks and let the water in. This is why drain tiles are used to direct the water to the sump pump and pump it away from your basement.

If you want it to stay dry dry all the time pros will solve it. I know you said that is not an option but honestly you are not going to have a dry basement unless they take care of the issue that is causing the wet basement which is rain leaking in the walls and through the floor. Trust me we JUST went through this. 2008 spring heavy rains, basement started to always be wet, we wet vaced twice a day and had 2 dehumidifiers running 24/7. 2009 we spent the whole year wet vacing every day (except in winter). 2010 we had enough of wasting hours a day on the basement that was endlessly wet and had mold growing in, Sept 2010 we called the pros. We have been dry since then, saved us from countless hours cleaning up after every rain. Yes it sets you back a little but we found our walls were starting to buckle in because of the heavy water pressure on the outside of the house (in a few years the basement could have caved in if we did nothing).

I do not work for any company dealing with water proofing BUT I have recently be through this. I even contemplated water proofing the basement myself... way too much work there and I did not know how to use a jackhammer to break the concrete basement floor :) You can only treat the symptoms for so long, to solve the issue look at the actaully problem.