No, dogs aren't allowed on the trail itself; it's a very sensitive natural area and an important resting/feeding stop for migrating birds. It's one of just a few places along the entire US west coast that arctic shorebirds return to annually, so it's important to keep it safe and welcoming for them.
Dogs and bikes on the airport road only, they are not allowed on the actual refuge boardwalk trail.
Because it is a national wildlife refuge, pets are prohibited https://www.fws.gov/refuge/Grays_Harbor/visit/rules_and_regulations.html
No dogs allowed. It is a resting and feeding place for shorebirds and raptors. I recommend enjoying the walk during high tide. It brings the wildlife close for viewing and photography.
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