How can I access a new client's pre-existing website?

You will get either (S)FTP or SSH access. Try to get the latter. Chances are that they have some sort of database admin tool like PhpMyAdmin, or a self-management-interface of the host that you could use.

However: As soon as you have access, you are responsible for not bringing the page offline by some typo or deleted files. Always, always, always make a backup as the very first thing that you do when you get credentials. Don't look around in the code, don't "just quickly change a color", nothing. Get a backup done! Then use that backup to create a development environment where you test and stage all changes, get customers approval, and THEN you put changes live.

Generally speaking, the site belongs to the client, not the developer.

Whenever I have taken over an existing website, the client has simply instructed the previous developer to hand all passwords and documentation to me.

So, yes, you ask the client, explaining why you need to know. The chances are they will have some idea of how the site was built, but they will want to ensure continuity and so will be on the side of the new developer.

A web browser is generally a good tool to use to access a website.

I wouldn't recommend using any kind of automated webcrawler or robot though. You can generally find out fairly easily whether they're using a CMS (Wordpress, Drupal etc) and which JS libraries, frameworks etc they're using. If they're large enough, they might outline their tech stack on stackshare.io.

If you're prospecting for work then why not identify usability issues to inform a presales pitchdeck or similar?