Is it permissible to pay someone money to expedite obtaining a spot in the vegetable market, and is that considered a bribe, and is the one who pays the money sinful?

What the questioner mentioned about paying money to someone to prioritize his application over those who preceded him is precisely bribery, because it is paying money to invalidate a right or uphold a falsehood. The precedence of those who applied earlier is their right, and paying money to usurp that right is considered bribery. It is one of the major sins, and its perpetrator is cast out from Allah's mercy. "The Messenger of Allah cursed the briber and the bribed in judgment," and "The briber and the bribed are in the Fire," and the mediator is also included with them.