You are outta luck, lazy-ass. And you will get little sympathy from most folks, I would suspect. However, I can empathize a bit and suggest ear plugs. If it is the scream of the two cycle motors on the blowers and the weeders that are waking you ear plugs do a pretty dang good job of blocking out the worst of that sort of noise.
Those motors are a nuisance I often find unacceptable myself. They are also eco-killers and put out as much pollution as an SUV- if not more. And they are responsible for the hearing loss of thousands of unfortunate landscapers.
Have you tried to just talk to the crew? They are on a weekly maintanance route and may be open to doing you a solid and hitting some other properties first before getting to the one next to you.
You could also talk to the property managment company that operates the building. While they may not wish to influence thier schedule you may be able to talk them into getting rid of the lawn and going with beds. Unlikely, but what the heck.
Ultimately, though, I'd have to side with the landscapers on this one. Spring and Fall is when landscapers make their nut and they need to bust ass to do so. They get up early and work hard all day.
I doubt that the entire crew is illegal and even if they were there are plenty of other day laborers that the company can hire if these get scared off.