provide programming solution for multiple sites using 1 DB

Project : Challenge: We have a mysql / php  site with a sub domain see URL in attached  We would like to have other websites integrate our site with their site almost like an I frame or we even replicate the site on their domain as their own subdomain with our site The problem: We want to have 1 database in our sites that control all other sites using their front end on different domains . Can you assist in developing a solution like this ? We can create an interface for all other participating sites so every user that registers from their end has a lifetime "stamp" in our one main DB almost like an affiliate that every user that they refer and register on our site or bringing other userswill carry that affiliate code . Please tell us your solution and budget to do it  the logo that we have should stay it just need to integrate with other websites in a smart way that user can register on another site and do all activities like on our site but the database is on our end  We are sure this can be done., but not sure how we will proceed with your solution if your solution make sense  environment is LAMP Thanks This project is just to provide a tech solution and then your estimate to do this for us