I'm planning a trip to Los Angeles in 6 weeks and have my eye on a specific star. Is it ethical to contact the several agencies she has ties with to see who is most responsive and gives me the best price? Any thoughts would be appreciated, thanks!

There is no problem in contacting the several agencies and ask them about rate and to see if you feel comfortable working with them.  

Be sure they are an agency with a good reputation on this board.  
If an unknown agencies offers you half the rate of the reputable ones be very leery of bait and switch.  

Hope you have fun here in LA

nothing wrong with shopping but you might also try and book direct with her. If she's on Twitter, which many PS are, you might try and contact her directly, tell her you can provide references, etc and see if she will book with you direct. You will probably save anywhere from $200-400. This has worked for me several times.

You're inquiring about ethics....seriously...it made me giggle....and I usually don't giggle...chuckle more. Do what you need to do to get your dick...I mean foot in the door.