If you are using a local ip as host for your connection there is no way that shinyapps.io can connect to it, you need to use a public ip that could be reached remotely, also, you may need to talk with your IT department to figure out if your Mysql server is configured to accept remote connections and what logging method is needed.
Yes, that is an option, you can upload your data to a sql server on a cloud computing service for example, also if you don't need to update your data very frequently you can export your data to a file (e.g. a .csv file), deploy this file along your app and read the data from it.