How to unlock 200cc mode

According to Mario Wiki:

Vehicle parts are unlocked by collecting coins from Grand Prix, VS Mode, and online races. Players can unlock new parts for every 50 coins that they collect. However, once players have collected 1000 coins, the coin amount needed to unlock new parts increases to 100.

Now there are 38 kart pieces that can be unlocked (not counting any golden items.) So doing some quick math:

1000/50 = 20

So that would leave 18 pieces that would need 100 coins to unlock, so:

100 * 18 = 1800

And put together:

1000 + 1800 = 2800 coins.

So with 2800 coins you'll have all kart customizations, minus the golden pieces which have special unlock criteria:

• Gold Standard (Kart): Obtain a minimum of one star ranking in all cups of every engine class.
• Gold Tires: Beat all the staff ghosts on each course.
• Gold Glider: Obtain 10,000 coins.

The last one in particular is of great interest, since it far out paces any other coin bases unlock criteria. It is impossible to unlock all of the other non-golden pieces without unlocking the Gold Glider, so in one sense the correct answer to your question would be 10,000 coins.

Summary

• 2800 coins to unlock all non-Golden pieces.
• 10,000 coins to unlock all pieces that can be unlocked via coins.
• Get a one star overall rank, and beat all staff ghosts to unlock the last two pieces.

On the IGN page you linked it says "Vehicle customization parts unlock randomly". I've confirmed that mine are not unlocked on the order of that page.