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According to Wikipedia, there are several places named Petropavlovsk in the huge country of Russia. No surprise.
Petropavlovsk-Kamchatsky is in Kamchatka and should be the correct Petropavlovsk.
I note that according to Google maps there is a body of water southwest of Petropavlovsk named Vilyuchinskaya Bukhta. That means Vilyuchinskaya Bay.
It would be logical for a Vilyuy river to enter that body of water but I can't see any rivers entering it on the map. I note that the Google Maps map of the area show some very narrow and very straight bodies of water that look like some kind of canals, but when I switch to the satellite view they look like creeks or rivers.
Thus I suspect that you may need better maps than I can find online, or historical maps.
If Khlebnikov received news of the shipwreck two days later, it should have happened within two days walk, ride, or sail of the location he was in when he heard the news. That certainly makes it seem like it was not too far away.