Airstream owners can be pretty stingy with letting folks rent out thier trailers...especially vintage ones in "immaculate condition". Not that they won't under certain circumstances (we have) but the more info you can share the more likely you will find what you are looking for. You may want to give some additional information as to what & how you will be using it.
See this thread: Would you lend out your Airstream?
Unfortunately, alot of people who want a vintage Airstream for a photo shoot, look at them as "props" and don't treat them with the same care as a fussy owner that knows
the trailer. We actually walked away with some damage after one of our photo shoots...nothing "major" to most, but an obsolete part was broken and insurance doesn't cover something like that and no amount of money can "fix it".
Have you checked with Iowa Boys
? They rent out trailers as a part of their business and are in the LA area...