Sometimes there's an R32 GT-R '89 that falls under the 10k price-point, but usually, it's at 11,000-16,000 in the used lot.
It depends on what you're trying to do: if you absolutely need a Kei car, then the Demio GL-X can be had for 4-8K, and will have exactly 98hp stock, the limit for the Kei races...yet you won't be stuck with a total wimp car once you tune it to 250hp.
If you need an FR car, then the non-GTr Skylines usually work, as does the '93 Supra turbo (one of the game's best buys!), and occasionally the '91 Twin Turbo drops down to under 10K. For light-weight battles, the '83 RX-7 GT Turbo is an absolute steal; fully-tuned it's got a low curb weight and plenty of power. And of course, you can't go wrong messing around with the 4 variants of AE86s/Truenos/Levins or the numerous Miatas.
For 4wd, the '91 Celica GT-Four and it's RC variant are nice. You can't touch Evos or Imprezas for under 12K...I've tried all the time, and just don't seem to get down that low in price. The Pulsar GTi-R, can be had for a nice price, too.
As for FR, there's nothing too exciting; the '94 FTO GPX gets low sometimes, but usually, it's north of 10K. But the '89 Mazda Protege (Interplay) is usually less than 5K, and you can get a lot for your money with that one. The '93 Civic Si is an okay buy for under 10K.
As for MR? Not much to choose from under 10K: the '86 MR2 and sometimes the '96 MR2 G-Limited can be found for under 10K.