This is a process we've been through on both sides a number of times. When a client has left us, it is because we, as an agency, had chose to end the relationship due to an abusive work environment/demands, or the classic: never paying ones' bills on time (if at all). We also pick up work a lot where a previous dev has left off, which can vary in terms of time and cost very, very much depending on your needs.

The first question that may solve this headache for you more immediately is: is your code base currently on GitHub or any other repository provider? This makes transferring source code exceptionally easy, and for our clients we create a new repository (always private, which you'll need a paid account for) for every new client project, whether it's an iOS app or a website.

If you DON'T have access to your code (and don't forget all of your creative assets and other strategy related documents that your previous provider may have created as a part of your project scope) then things will just be a tiny bit messier (and costlier) during the transition.

While it's true that many developers really need to start from scratch on certain transfer projects like these, if the previous company you were working with was any good (charging at least $90-$150/hr) then the code that they wrote likely adhered to a certain set of standards and best practices.

Any responsible web agency will ensure their code can be taken over at any time by a new dev. This is helpful for everyone involved, always, in the short and long term.

Anyways, if you'd like to share a few further details about your project I'd be happy to chat more :)

Cheers,
Lara