Hosted by 3rd parties
There are several third parties where you can run a Firefly III instance.
Synology
Mariushosting.com wrote an excellent guide on how to host Firefly III on your Synology NAS.
Heroku
Firefly III supports Heroku. You can deploy Firefly III in Heroku after you register for an account.
Considerations when using Heroku
Heroku uses what is called an "ephemeral file system" and it will not be able to store attachments. They will be deleted every day. Don't use Firefly III on Heroku in combination with sensitive or rare file attachments.
Softaculous
Warning
Your hosting provider may not be compatible with Firefly III, even when it's listed in your Softaculous application list. I cannot support you with this.
Firefly III is featured in Softaculous. If your (hosting) server provides packages using Softaculous, Firefly III will be available as a package there. They even made a special demo site.