This is what I do:
A. I look that site's source first. I validate it, see what's going on with the CSS, etc. All I need to know is in the source. Even if the content is generated server side, the client has no knowledge of that fact, whether the client is a browser, bot, phone of fridge.

B. If the source needs a lot of help, then I go back to the client and ask how the content is generated. I also tell the client WHAT I found that needs to be fixed. At this point, we know what needs to be fixed, and just need to know how to fix it. Now, if the client does not know how the content is generated, then I ask for the developer's contact information, and talk to the developer. See, I still have had no need to access the client's site.

At this point, once I know what's wrong, and how to go about fixing it, I can give a quote - but I never needed to have access to the client's site.

My suggestion to you is to Google for the company who is offering you services (NOT by using their company name). If they do not appear quickly, then I would wonder what kind of services they are REALLY offering.