You are certainly free to use shinyapps.io's free plan in a technical sense, that is, you can deploy Shiny apps there. But the answer to your question depends on your organization's policies.
For example, I would be nervous to expose any personal, proprietary, or protected data with a Shiny app deployed to the free tier of shinayapps.io. At least until your IT org has had a chance to review things. On the other hand, many apps only deal with publicly assessable data. But as innocuous as that sounds, whether or not you can expose that to shinyapps.io again probably depends on your org's policies.
One good avenue might be to check out RStudio Connect. It's on-prem software that let's you deploy Shiny apps (also apis, reports, data, and more). You can use it to develop, test, deploy and maintain this kind of work.