Sungura,
If you're looking to go "cheap" with $100 budget, then your options are severely limited. You can buy a "cheap" point-and-shoot for $100, but it won't have the features you desire. You are going to get an on-camera flash and a lens that pops out. But you'll be able to shoot in the AUTO mode and won't have to worry about setting shutter speeds of f-stops. In dry caves you can get good photos, but in humid caves you are going to have problems with fog. You might be able to find something that cheap on the second-hand market that meets your caving needs. eBay?
You can get really good cave photos (even with a simple point-and-shoot (P&S) camera) if you can get the flash away from the camera. But using an off-camera flash starts to complicate the whole process of photography.
The simplest (but not necessarily the cheapest) set-up would to use a camera with a hot shoe, and a hand-held strobe. (You can buy used strobes very cheaply.) In order to shoot in AUTO mode, you'll need to run an electrical cable (not a fiberoptic cable), also called a sync cord, between the camera's hot shoe and the strobe, so that the camera and strobe have direct communication. Most small, cheap P&S camera do NOT have a hot shoe, so you'll need to purchase either a cheap used camera with a hot shoe or a newer, expensive P&S camera with a hot shoe. Your cable will need to be specific for the camera and the strobe. Be warned that there are a few different types of connections between the strobes and hot shoes, such as one, two or three electrical connections.
A more advanced point-and-shoot with a hot shoe
A sync cord that connects a camera to a strobe
Once you eliminate the cable between the camera and strobe, the camera no longer knows that is should automatically expose for a flash. So you are going to need to shoot in manual mode and set the shutter speed and f-stop. You will need some kind of optical slave (not so good) or a radio slave (very good) to get the camera to trigger the strobe.
The problem with using an optical slave is that the camera's flash will still fire and you'll still have problems with fog. Most people will cover the camera's flash with something that blocks visible light (to avoid the fog) but still allows for infra-red light (to trigger the slave). I cut a small rectangle out of a white plastic shopping bag and cover the strobe using duct tape, leaving only a small sliver of light to penetrate to trigger the strobes.
Pentax has an infra-red slave system with their digital SLRs that allows you to shoot in AUTO with multiple off-camera Pentax strobes that are equipped with infra-red receivers. This is not cheap, but is very convenient because you can shoot in AUTO even with mulitple strobes, and you have full control of each remote strobe from the back of the camera!