yes, they do. If you feel comfortable soldering a chip by yourself, you could get a modchip for $40. If you don't, you could get a chip and installation service for $100 or so.
There are also fliptops - it's a replacement cover for the PS2 that lets you take a CD out without using the eject button. You use this with a disc like Swap Magic, which boots up then prompts you to swap discs using the cover. (pressing the reject button makes the PS2 re-read the region code of the disc, making the swap disc unusable)
However, since the swapping method takes time, you'd probably prefer to get a US PS2 and only swap for Japanese games.
There are even more considerations, though. A mod chip would boot all games, but PS1 games (from both regions, I think) have extra mod detection that you need to use gameshark codes to eliminate. Swap discs only play PS2 games. You need a Gameshark to play PS1 games, and that requires going through the menus and enabling both ignoring the region code and the anti-mod-detection codes.
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this thread has helped me out a lot)