I'm a PhD student and hope to get my thesis defended during the next winter or spring. Before that time, I want to apply for a postdoc. In the lab which is my main dream (though I think it is very hard to get there), there are some permanently open positions for postdocs, and candidates are encouraged to contact the Lab supervisor by e-mail, without any further specifications on their web-site. This lab is in the US. I will certainly send in this e-mail my CV and cover letter, but how would you recommend, is it better to ask my current supervisors and co-authors to give me recommendation letters, so that I could send them at once, or to give their contacts in this first e-mail to my potential new supervisor? Thank you in advance.
The request to contact by email (without specific requirements) suggests a less formal application process.
I would send a brief email (a few sentences to a couple of paragraphs) explaining your interest in working there and why you would make a good candidate. The lab supervisor is probably very busy, so by keeping it brief you have the best chance that it will be read.
It would also make sense to attach your CV, so they have a sense of your skills and background.
However, I wouldn't include any other materials (such as a separate cover letter or letters of recommendation) at this stage. Wait until they express interest in continuing the conversation (and then be led by them about the next steps).
Exception: if one of your referees knows someone at the lab.
In this case, I would ask the referee if they would be willing to make an introduction for you. This is likely to make more of an impact than you contacting the lab "cold".
Even if the person the referee knows is not the same as the contact person for the job, having the referee contact the person they know is probably the best way to go.