You may not have the original files to upload, but you can certainly download the files that are currently on the system to your PC. You can use an FTP client or the hosting control panel (LPCP, cPanel, et al.) often will be able to do it. Depending on what the original source was for the files and how they were produced, it may be easy to update them (e.g., PHP source -- everything you need is on the site) or difficult (e.g., Dreamweaver -- control files are on a PC). Do you have the password to access the control panel or use FTP? If not, open a support ticket to have the password changed. You will have to prove ownership of the account.
It was very unprofessional for your "colleague" to have up and left without turning over the site source and instructions to you, unless he was abruptly fired for no good reason (in which case, your management takes the blame).