The OED states that the verb "impinge" is intransitive. However, example sentences such as the following have (as I see it) the verb taking a direct object (highlighted in bold):
Several factors impinge on market efficiency.
The proposed fencing would impinge on a public bridleway.
In the above sentences, is the verb not transitive?
No, it would not be classified as transitive—at least, not according to the Cambridge Grammar of the English Language's definition of transitivity.
A transitive verb, such as crush, takes a noun phrase called the "direct object" as a complement. (E.g. noun phrase "the lemon" in "I crushed the lemon" is the direct object of the verb crush.)
In the sentences you give, impinge takes a prepositional phrase headed by the preposition on as its complement. That prepositional phrase contains a noun phrase (sometimes called the "object of the preposition"), but the fact that this noun phrase is embedded in a prepositional phrase makes it not count as a direct object of the verb.
Both transitive and intransitive verbs can take complements, but of different types. Only verbs like crush that take noun phrases as complements are categorized as transitive verbs. Impinge would be transitive if you could say things like "Several factors impinge market efficiency." (The comments indicate that the OED records transitive uses of impinge.)
Related previous posts:
How is transitivity defined in CGEL?
Brett Reynolds' answer says:
For CGEL, transitive means specifically 'taking a object as complement'. Moreover, I recall personal discussions with Pullum and Huddleston in which they confirm that only NPs function as objects. Verbs or prepositions which take other types of complements, such as clausal complements, predicative complements, etc, are not transitive.
Araucaria's answer says
An object is a special type of complement of a verb. [...] It is the preposition heading the preposition phrase which has a special relationship with the verb apologise. So because someone isn't a complement of the verb here, it cannot be considered a direct object.
call on - trans or intransitive verb
BillJ's answer says
A verb is only transitive if it has a direct object in the clause that contains it. [...] Note that complements of prepositions are not direct or indirect objects.
Therefore, the only way "impinge on something" could be transitive according to the CGEL definition of transitivity would be if "on something" did not constitute a prepositional phrase. I'm pretty inclined to say that "on market efficiency" and "on a public bridleway" are prepositional phrases, but some of the comments have suggested that the structure of your examples is "[impinge on] [noun phrase]" instead of "[impinge] [on noun phrase]". This issue is discussed in the following places:
Greg Lee's answer to "Transitive / intransitive verbs" discusses that topic.
This handout from a course taught by Bert Vaux and David Willis, professors at the University of Cambridge: "Li8 Structure of English: Verbal syntax and complementation"
Another related post:
I agree with you, those examples clearly show transitive uses of the verb impinge. The verb takes two arguments, a subject and a prepositional phrase. If a verb takes two arguments then it's transitive. In fact I can't think of any intransitive examples, unless you would count a clause with an elided argument (and I wouldn't, elision is a process that changes the final output, but doesn't change the natural valency of a verb IMO.):
John: One factor impinges on market efficiency.
Mark: No, many factors impinge.
