Oh Lord No!
You do not want access to the live site lest you make a mistake and are able to take responsibility for anything that could possibly go wrong even if it has nothing to do with you. It may be necessary that you deploy the site for the client. However, in this case, it seems that the client is willing to take on the this task.
To answer your question directly:
You must have a proper working representation of the site environment in order to do your work and not rely upon the site environment to perform development upon. For this you will need enough details of the existing environment in order to properly replicate it for your work. This is so important that you must require it before work can really begin. The good news is that this should be relatively easily achievable.
The longer version:
In a professional/commercial environment, developers will never be given access to the server, site, computer room, and in some cases even the floor or building that a site is hosted in. This is standard security practices designed primarily to control systems stability more than anything.
It is expected that development create and maintain an environment that matches the production environment. In some cases, production will do this for development using sanctioned build procedures and stored and protected images and product installs. In some cases, development never owns any hardware or installs any software, but does finance it within a development environment managed by the production staff primarily or at least overseeing it.
Since the days of hosting, and more in particular recently, this discipline has been disappearing largely due to overly casual relationships and smaller one-on-one development efforts outside of traditional development/production environment. Some flexibility is required but not so much that the outcome is jeopardized.
Having said that:
It is expected that you should have some base information by which you can build an accurate development environment. If this is not available, I would walk away until it is. Most environments are easily replicated within a reasonable limit. For Windows environments, this is not as easy as in the Linux environment and may cost real money that you will have to consider. However, for most Linux environments, this is easily achievable. In your case, you are using WAMP which can mean a large variation in how the operating system is installed and how the software is installed. This may not matter in your case, however, in production environments, every tiny component of the OS and install base is evaluated to ensure that both development and production are fully in sync and secure and that development can properly match the production environment. Even a small .dll version difference can make or break a system.
Part of the development/production standard is that development will provide technical requirements that may not be stated in the original requirements given by the client. It is expected that the client is not technical and that the development staff translates the clients requirements into a technical requirement and feasibility study and come up with a working design and cost analysis. Production then can take these development requirements and evaluate it for acceptability within a secure production environment. In addition, development must develop as part of their work, a deployment procedure that is also accepted by production as being safe for the environment.
In more casual development relationships, the developer must not only understand the existing environment, properly replicate it, create a design specification, project the feasibility and cost requirement of developing such a project, but also take on some of the responsibility of what would be the production role in that they must understand whether the product they are proposing is secure and stable and can assure it within a reasonable limitation. In addition, the developer should have a proper plan for deploying the website even if the website is being deployed by the developer.
This can be a simple matter of course and often should be. However, as professionals, we must be ready to backup our work in a professional manner especially in the case where the client is being unreasonable or expectations do not align.
In other words; CYA.