I'm working on a tool which manages WordPress instances using puppet. The flow is the following: the user adds the data of the new WordPress installation in the web interface and then that web interface is supposed to send a message to the puppet master to tell it to deploy it to the selected machine.
Currently the setup is done via a manifest file which contains the declaration of all WordPress instances, and that is applied manually via puppet apply on the puppet agent. This brings me to my 2 questions:
To answer your first question, yes there's absolutely a way of doing this via a puppetmaster, what you have at the moment is a masterless setup which assumes you're distributing your configuration with some kind of version control (like git) or manual process. This is a totally legitimate way of doing things if you don't want a centralized master.
If you want to use a master, you'll need to drop your manifest in the $modulepath of your master (it varies depending on your version, you can find it using puppet config print modulepath on your master) and then point the puppet agent at the master.
If you want to go down the master route, I'd suggest following the puppet documentation which will help you get started.
The second question brings me on to a philosphical argument of 'is this really want you want to do?'
Puppet traditionally (in my opinion) is a declarative config management tool that is designed to make your systems look a certain way. You write code to determine 'this is how I want it to look' and Puppet will converge to make it look that way. What you're looking to do is more of an orchestration task (ie when X do Y). There are ways of doing this with Puppet like using mcollective (to trigger a puppet run) which is managed by a webhook, but I think there are better tools for the job.
I'd suggest looking at ansible, saltstack or Chef's knife tool to do deploys like this.