In the vast majority of cases, no, but you should check the license.
Normally, when you buy a game, it's copyrighted. You can't re-upload or distribute the game, or modify the source code. Doing so violates copyright laws and is illegal; you could be hit with a C&D or DMCA.
Some people offer content under other types of licenses, such as the MIT license, and in those cases, you can redistribute the game. So you need to consider each case individually, and it's best to ask the game's creator directly before purchasing. Itch doesn't force you to use any type of license, so the license used depends on the original creator of the content and not the platform itself.