Greetings,
When you dont have access to source code, at that time only one option we can think of which is getting all the information you can get and building it fresh with all the required functionalities.
So please share site url & access whichever you have with you.
Also as you said you have db dump, if it is whole db dump, there are better chance of that being useful.
We have 15 years of experience on LAMP platform and we will help you best in this migration/redevelopment.
Thank you,
CWE