PHP is a turing complete language, and MySQL is a relational DB, so between the two there is very little you can't accomplish, unless you want to start doing full text search against millions of records in a database - at which point you need to augment the stack with something like ElasticSearch, or switch from MySQL to Postgres.
If it's a traditional request/response site, or a simple API to power a rich client-side app or mobile app, yes, absolutely. In fact, I wouldn't choose any other language for this since PHP gives you great architectural design features like object type safety, but without the need to for a manual compile step. So you get similar benefits of Java or C#, but with the instant interpreted language feedback of Ruby and Python. And because of the stateless nature of PHP, you don't have to restart a server like you do in Python. Have an error? Correct it and refresh the page. Done (unless you switch over to a looped process with PHP PM, but that's not standard PHP).
However, if it's a real-time app with live growl/push notifications, chat, and all sorts of other things, then no - you cannot really do this with your requirement of plain PHP + MySQL alone. PHP must be either made to run in a loop with a runtime like PHP PM or react PHP, or you must augment the stack with Node + Socket.io + Redis.