I started negotiations with a dealer on a used car. The next day someone else did and they came to an agreement, but didn’t do anything formal. Am I allowed to now attempt to submit a higher bid? Would that be a violation of Ani Hamehapech or anything else?
This is possibly a case of ani hamehapech.
The Gemara’s case of ani hamehapech is a similar case of somebody who negotiates a purchase after the seller had already reached an agreement with another buyer (see Kiddushin 59).
Although you started negotiating, I understand that you didn’t reach agreement, which means that it was permitted for somebody else to negotiate. Once he has reached an agreement on price and terms, it will not be permitted for somebody else to compete.
However, if the purchase is a unique item, and it will be hard to find it elsewhere (this is often the case for land acquisitions), Rabbeinu Tam rules that the idea of ani hamehapech does not apply, and this is cited by the Rema (Choshen Mishpat 237).
Good luck and best wishes.