I recently had an oil tank removed from under my driveway. The disturbance has caused water to POUR through some cracks in my basement during heavy rains. I'm getting a good company to fix the problem from the outside but it's going to be a week before they can start. Can I do something temporarily?
It sounds like you are already aware that the best way to keep water from entering your basement is a good exterior waterproofing system. There are a few temporary measures that can be tried on the inside, but if you have a lot of rain, they will probably not completely prevent moisture intrusion.
When you say that you have a cracked foundation wall, are you seeing hairline breaks in the concrete or are these actual splits that are 1/4 inch or wider? If they are just hairline width, you might try using some concrete caulk - it's available at just about any home improvement store. There should be some specifically marked for concrete repairs.
However, if your foundation is showing splits larger than 1/4 inch, caulking alone may not suffice. You might try wedging some caulking backing rod into the cracks prior to applying any caulking. Backing rod is a foam type product that painters use to fill cracks that are too large for just caulking. It's available in several sizes and can be found at many home improvement stores and just about any paint retailer.
If you can't find backing rod, expanding foam insulation spray might also be worth a try. Most varieties aren't waterproof, but they might act as a deterrent until the exterior waterproofing is done. They expand when applied so should do a good job of filling the cracks.
I'm not sure if you have already checked on it, but there are some waterproofing systems that can be applied on the interior of the foundation walls. They normally involve shooting an epoxy type material into the cracks using a special tool. There are contractors who specialize in these types of repairs. I prefer the exterior waterproofing methods, but sometimes they can be very expensive -- especially if you have a lot of landscaping around your home or a driveway needs to be removed.
Whichever permanent method you choose, make sure you use an experienced waterproofing company that warranties their work. The last thing you want is to spend a lot of money repairing the cracks only to have them start leaking again the next time it rains -- properly waterproofing a damaged foundation wall takes a lot of skill.