What's the difference between the two following sentences?
a) I came by subway in order to not be late.
b) I came by subway in order to not being late.
For my non native English speaker ears, both sound natural. Isn't it?
"in order to not being" is ungrammatical, because "in order to" always takes the plain form of the verb. So "in order to not be late" is correct.
English treats the subway (or any vehicle) as a means of travel, not a tool, so the preposition is by, not with: "I came by subway" (or "by the subway", if you want to specify a particular subway, but that's less common).
So the sentence should be "I came by subway in order to not be late." (It could also be simplified to "I came by subway to not be late.")