Concerning seeing an abortion specifically, it isn't something that one would typically "shadow," and you wouldn't be able to see much even if you were present in the room. Many abortions are medical and do not involve any procedures beyond giving pills. Most others are done endovaginally (going in through the vagina and dilating the cervix, then removing the fetus with suction), in which case you wouldn't be able to see what was going on anyway. Even fairly large fetuses can be removed this way (you may recall the debate over so-called partial birth abortions). In other words, it typically isn't an open surgical procedure where you could scrub in and observe what was happening like you would in a typical OR.
If you're interested in reproductive health, you should look to volunteer or shadow in that venue. I know of several people who have volunteered with Planned Parenthood, for example. It may be a little harder to gain women's trust if you're a male, but there are plenty of male OB/gyns, so it's obviously not impossible.
I also don't think it's abnormal to just be curious about how abortions are performed. If this is your main motivation, I suggest contacting your local Planned Parenthood and asking if it would be possible to visit to learn more about abortions. When I was in medical school, we were given the opportunity to visit a local center that offered abortions, and they showed us all of the instruments and how the procedures were done using models. This was done after hours when there were no patients in the facility, so it did not violate anyone's privacy.