I am trying to achieve the Sportsmanship badge. I wonder if it's possible to know how many times I upvoted competing answers until now?
I wonder if it's possible to know how many times I upvoted competing answers until now
No, it's not possible. As of April 2015, this is possible using the features of the new profile page, as described in this other answer.
Technically speaking you can check such data at Data Explorer. (However, the data gets updated weekly, so it won't be 100% accurate.)
How close am I to Sportsmanship badge?
This query returns all your answers, where you upvoted the competitor.
You can see the total values of the returns, at the bottom of the table.
You only need your userid. You can acquire your userid from the URL of your profile page.
With the introduction of new "activity" tab of profile page, it is now possible (but this new page is only tested on MSE right now):
You can now select "Sportsmanship" by clicking on the pencil and choosing the badge.