Yes you can use a service like Dropbox or iCloud to do that for you.
Assuming you have iCloud:
- Create a new TaskPaper document
- Choose Save, and In the save dialogue choose one of the iCloud locations
- Once you’ve saved to that location you should be able to go to another computer that’s logged into the same iCloud account and open your document from that same location.
Depending on how much sense that all makes you might want to do some general “How do I use iCloud Drive” searches to learn more. Here’s the first result that I came up with: