Yes, your client should highly consider upgrading this website/application to run on php 5.6 and changing hosts if the current host does not provide boxes with php 5.6. Running a production box with php 5.3 has severe security and technological limitations that increase risk and cost of development significantly. I’m going to assume your host is also running an out of date, unsupported os as well that is susceptible to security vulnerabilities at the server level. That being said I understand the politics and work involved in doing all this. Depending on the website/application size and dependencies it can be a large task. Especially if you are running a cms and/or framework version which does not support 5.6. In that scenario you effectively need to upgrade the cms/framework to a newer version which can be difficult without migration paths for the core and dependencies. Worse case scenario the project needs to be rebuilt. Best case scenario is the project is small enough that moving to 5.6 is seamless. I think most people end up somewhere in-between those two extremes.