The best and quickest way to make 10 million yen is gambling at the casino. You can unlock the casino once you arrive at Kamurocho around chapter 5. There should be some punks squatting down in the north streets of Kamurocho which will trigger a sub-story. Simply finish that sub-story and you'll be allowed access to Dragon Palace which contains a casino. The other Casino is available in Chapter 8 when you get access to purgatory via the story. I would recommend the purgatory casino as the save box is closer in case you lose. Also you'll need 100k yen to purchase the initial chips. Anyway, on to the ways to earn money at these casinos...
Roulette: My preferred method. The game gives you three cheat items; zero jewel (50/50 chance 0 or 00), red jewel (100% chance Red), black jewel (100% chance Black). You can get a zero jewel from Rikiya if you showed him the sights during chapter 6, and the black and red jewels can be found from coin lockers. There are only one of each in the entire game so don't waste them. Onto strategies...
Zero jewel is self explanatory, just bet 1000 chips on either 0 or 00 and reload your save if you lose. Red jewel and black jewels depend on how lucky you're feeling, if you bet on a single red or black number (depending on which jewel you use), you have a 1/18 chance of winning. But if you put 1000 chips in the middle of two same coloured numbers you have a 1/9 chance if either number comes up. Either way you get 36k or 18k chips for winning after betting 1k.
Blackjack: The other preferred method. There are two cheat items you should use; Bust Amulet and Blackjack Amulet (acquired through coin lockers). First use the bust amulet to force the dealer to constantly bust 5 times in a row, this will get you on a winning streak for higher bet amounts. Once the dealer has busted 5 times, use the blackjack amulet to guarantee 5 blackjacks in a row. This will ensure you get a massive amount of chips.
Poker: Don't bother unless you love poker. The winnings are pathetic. You could get 10k winnings in about 15-20 minutes but it's not worth the trouble as it's so slow.
Use whatever method you like best above and be sure to reload your save game if you lose (that's why purgatory is better because the save box is closer).
Once you've acquired around 100k tokens (2-3 wins from Roulette and some Blackjack would do it), buy however many platinum medals you need from the prize exchange. They cost 1.5k chips each and sell for 150k yen at the pawn shop. Also note the 10 million has to be on you at the same time, so you can't get 5 million, spend it, then get another 5 million. If you're coming up short, you could sell all your stuff to get over 10 million, then reload a save before selling everything.