It’s not quite clear if a wild owl can deliver messages.

It’s not clear where Sirius got the owl he used to send his first letter to Harry. Sirius said the owl was the best he could find and was eager, but that’s all he mentioned.

“I won’t tell you where, in case this falls into the wrong hands. I have some doubt about the owl’s reliability, but he is the best I could find, and he did seem eager for the job.”
_{^{- Harry Potter and the Prisoner of Azkaban, Chapter 22 (Owl Post Again)}}

This could mean that he needed an owl that was trained to deliver mail but doesn’t confirm it for sure.

Sirius did use other types of birds while in hiding instead of owls.

When Sirius is hiding in some tropical country, he uses tropical birds rather than owls to send Harry letters.

“Harry had received two letters from Sirius since he had been back at Privet Drive. Both had been delivered, not by owls (as was usual with wizards) but by large, brightly coloured, tropical birds.”
_{^{- Harry Potter and the Goblet of Fire, Chapter 2 (The Scar)}}

It’s unclear whether those birds were trained by wizards in the region to carry mail, but it does show owls aren’t the only birds that can carry mail.

A JKR Pottermore writing says most owls belong to wizards anyway.

In a Pottermore writing, JKR gives a bit more detail on how owls carry letters, and mentioned that most owls either belong to a specific wizard or their country’s Owl Postal Service. Presuming this is true, it’s likely then that Pigwidgeon was an owl who’d already learned to carry mail.

So numerous are the owls employed by wizards worldwide that it is generally safe to assume that virtually all of them are either the property of the Owl Postal Service of their country, or of an individual witch or wizard.
_{^{Owls (Pottermore)}}

The same writing also refers to wizards who train owls, and also mentioned that trained owls are expensive.

While the process remains mysterious even to those who train up owlets to become wizarding pets or postal owls, the birds appear to be able to make such a connection between the name and its possessor that enables them to trace the witch or wizard concerned wherever he or she may be.

However, it also says that owls learn quickly and enjoy delivering mail. So, theoretically, it may then be possible to train a “wild” owl to deliver mail.

Whether because they possess an innate bent for magic (just as pigs are reputed to be innately non-magical), or because generations of their ancestors have been domesticated and trained by wizards and they have inherited the traits that make this easy, owls learn very quickly, and seem to thrive on their task of tracing and tracking the witch or wizard for whom their letters are intended