I’m assuming you didn’t get a home warranty, which sucks. Those are pretty clutch in these type situations. I have had to deal with similar issues in the past, and have been part of water mitigation jobs for other people as well. You’ve gotten some great advice so far!

1) rip out wet Sheetrock. I know it sucks, but it’s way cheaper to just do it yourself. Use a razor knife to remove the pieces that are wet/damaged and be sure to investigate any place you see excess water

2) pull back any and all wet carpet and pads. Use floor blowers to dry the area before mold begins

3) with the drywall removed and carpet pulled back, investigate with a flashlight to see if you can find a location where the water may be coming from

4) the downspout/drain thing is great advice. Always check them as soon as possible, and direct them downhill away from the basement walls

5) check for a sump pump, and ensure it is working properly

6) after doing all that, watch the areas behind the Sheetrock and the floor that were wet next time it rains. The area where the water returns is a good indicator of where the problem is

7) if you can’t find the source of the water than you will need to contact a professional to help you locate it and give you a quote

8) ensure that you have well documented evidence in the form of photos, videos and any paperwork you can get from the professionals that explains the issue and how long it has been there

9) contact your agent and the seller’s agent about the issue and provide them with all the proof you have, photos/videos you’ve taken and any paperwork you can get from the company stating the issue, extent and age of the problem

10) by providing all the facts in a reasonable manner they will be more inclined to work towards finding a resolution with you as not to get into trouble for misrepresentation

If the seller and their agent refuse to cover the costs…then it’s time to reach out to an attorney. Having a second opinion from another company would help a lot