In Oregon, it's against the law to hitchhike even without your child. See ORS 814.080.
Specifically, what's illegal is standing on a roadway for the purposes of soliciting a ride. Getting rides from strangers, from the internet or whatever, is not illegal, it's just being on the road to solicit rides that's prohibited. So if you want to take a camping trip with your child, arrange your rides in advance.
(Note that this applies to Oregon law. If you're going through other states, their laws would apply while you're there.)
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