If you are independent vendor you have 2 possibilities.
First, take computer, connect it to network, install tomcat and mysql on it, deploy your web application, buy fixed IP, buy domain name and enjoy.
Second way is find hosting company that hosts java applications and provides mysql service. There are a lot of such companies. If you choose one you pay them money and they do everything for you. I believe that if you need java and mysql you will not find free hosting company. But try. Probably there are such companies that will provide service for free and add a couple of their banners to your site.
And yet another way is to buy virtual machine from for example Amazone cloud. You can buy the simplest linux machine. Install java+tomcat+mysql on it. Buy domain name and you are done. It is not free but it costs several cents an hour, so probably this will be the best way for you.