Short answer: yes it's possible. Longer answer: it will probably require some custom development during your implementation to get in place what you're looking for.
Sprout Forms has a Tags Field in the Relations Field section of the Form Builder. On the CP side, it's just a wrapper around Craft's Tag Relations field and will work the same way for Content Admins.
By default, this field just outputs a list of tags in a dropdown for a user to choose from.
But, does the field support submitting new tags? I don't know! As the Form is just receiving POST data I'd start by taking a look at how Craft's Tags field is sending new Tags when entries are saved in the back end and try to update the front end tag input to submit data in the same way and see if it works.
If it doesn't work, then we could consider that a Feature Request or you could address the problem by building out a custom Tags Field that behaves as you need it to using the Form Field API. You're more than welcome to use Sprout Form's Tags Field as a starting point.