"If you vote for me, I will <political promise here>" sounds like an oral contract to me, since it involves consideration in the form of a vote.
Oral contracts when done before witnesses may be enforceable in some jurisdictions, except when the type of contract in question explicitly requires a written counterpart, such as in the conveyance of property.
I'm curious if this could be solved at the federal level via checks and balances provided at the state level. i.e. could a State legislate that it is illegal for candidate for US federal office to make false campaign claims when in that state. e.g. Any promises conveyed in speech made by a US presidential candidate when campaigning in that state, would constitute a legally binding contract between that candidate and any constituent of that state that votes for that candidate if that candidate is successfully elected into office.