Our club does autocrosses, we used to do two a year, and now we are doing about 4-5, we get our insurance through the national club we are affiliated with, contact me directly if interested, I know it has all been cleared with them as kosher, but I don't want to spoil the party by everybody jumping in the pool. We do ours at the local school, there have actually been lawsuits that if you let one outside group use school grounds for something you have to let everybody use it or you are discriminating, so we use the high school parking lot on Sundays. I used to set them up through the school administration (somebody else does it now) and I was always concerned they would freak out about liability (the do require evidence of insurance) but when I would explain the area would be closed off to the public (I sort of hem hawed around, not dangerous, but we take all precautions just in case) their main concern was always "oh my gosh if you close it off we will other people be able to get to the school for other activities" not the concern I was expecting.
We are a multi marque all British club, but some Miata people and others who run SCCA stop by with their modern stuff, they can run if they join the club (and hence are covered by insurance) we usually end up half modern and half vintage british, and it is all good fun, as some of the modified Sprites and a couple other fast cars are very competitive with the modern stuff. Great fun, I did a track day a couple years back and though it was relatively boring, although I am sure competitive racing would be a whole nother kettle of fish, but autocrosses are boom boom boom from one turn to the next, always turning or setting up for a turn, and you only go one lap or two at a time, so it makes sense to drive it on the ragged edge, total concentration, and take your chances at the edge of the envelope to get a good time. Also still pretty safe as you are not fender to fender, and the only thing you will hit is rubber cones.