Full answer after the jump...
Skyward is such a robust and secure system that it will not allow external applications to input any data. This would leave the system open to exploitation and potentially sensitive data could be changed (maliciously, or otherwise).
However, thanks to updates to Google Classroom, there is a neat workaround!
Once you have completed grading your submissions, you can hit the "Download" button:
This will download an Excel spreadsheet of your class roster and the grades that they received. If the list of students matches your class list in Skyward, then you can simply copy the "Grade" column (just the scores) in the spreadsheet and paste it into the grading window in Skyward.
Not a perfect solution, but one that will save you a few precious minutes on every assignment.