You probably can't meet this client's needs in a reliable way. The results of this "integration" will probably end up being Worse Than Failure™. If you proceed with this job, please be sure you arrange to bill your client hourly for maintenance. There's no obvious reason for you to take a vow of poverty to support this ill-conceived scheme.
If the Access database (the mdb file) is relatively static, you can write a bunch of application specific code to sync its tables to a real RDMS like MySQL. If you do this in PHP, you'll use the ODBC driver. http://php.net/manual/en/function.odbc-connect.php
The machine on which the php sync program runs will have to have Windows-style CIFS or SMB file-system access (not, repeat not, FTP access) to the mdb file. You'll need to make the mdb file available in a network share, for that to work.
This shared-access to mdb setup is a potential source of unreliability. You can mitigate some of that by only reading, and never writing, the mdb file from your php program. You can also allow for recovery by making sure you have solid backups on the mdb file.
If your php sync program, and indeed any program touching the mdb file, can be run in a Windows environment, it will help maintain the mdb file's integrity.
Once you've pulled data from the mdb file using ODBC and put it into an RDMS, you can create your web site normally.
Another alternative: hybrid solution. Read application data from the mdb file, and read/write web-site data (passwords, sessions, and the like) to the RDMS.
It may make sense to try to convince your client to migrate from access to a modestly-sized MS SQL Server database. That's a fairly well-travelled path. If you do that, you can connect your php program directly to SQL Server.
This project sounds like your client saying, "I want to take all my customers to New York City. I have this bicycle. Can you drive it and carry them all?"