“On accident” is American English, as a British English speaker I’d consider it a grammatical mistake. The same goes with “I forgot it at home” and similar constructs. However they’re correct American English.
Another example is the use of "Write" in the sentence, "I wrote them". This is completely wrong to the British ear, which would be "I wrote to them".
A better example would have been "I wrote Alice last week". Correct US English, utterly grating to British English. ( Technically still might not grate if your brain jumps to Alice being a Poem or other work of art! )
Because in British English we write letters, we don't write people. I don't know the term for it, it's not transitive vs intransitive, it's the verb object having a different restriction.
"There was a problem at the mill so I wrote them" - this sounds wrong
"The mill workers were complaining so I wrote them a letter" - this is fine
"The mill workers were complaining so I wrote to them" - this is also fine
Mallory: "The mill received a series of letters. Those letters are evidence. Now, we're not leaving here until I find out who wrote those letters. Alice, did you write them? Bob, did you?"
Carol: "I wrote them."
As a non-native speaker, I find the sentence equally disconcerting, but it leaves me wondering what one would use to say something to that effect.
Really, "I forgot it at home" is short for "I forgot to bring it; I left it at home".