Is it possible for a Reform Jew to violate a mitzva in the Torah or break halacha, like breaking shabbos by doing melacha ('labor')? After all, from what I understand, Reform Jews don't believe in a requirement to keep mitzvos; thus, I don't understand how a Reform Jew can violate something there's no requirement to keep. Can he? How does that work?
Yes, a Reform Jew can absolutely transgress a mitzvah. That Reform doesn't say up front "here are all the mitzvot you must accept" does not mean that no Reform Jew accepts any. Reform (as taught today; I can't speak to early history) isn't about rejecting mitzvot.
A Reform Jew reaches an understanding of halacha through a different path than others, and some may never reach it at all, but one who does is exactly as liable for it as any other Jew. If I drive on Shabbat I am guilty of violating Shabbat; I don't get to say "I'm Reform so I don't have to do that".
The difference is that for any random Reform Jew, we don't know what his understanding of Shabbat is. So "is it possible to violate Shabbat?" Yes. "Is that guy over there driving his car violating Shabbat?" Yes, per halacha. "Does that guy over there driving his car believe he is violating Shabbat?" That depends on what he has accepted. If "that guy" is Orthodox, on the other hand (and not uneducated etc), then we can presume that he agrees he is transgressing. (Assuming in all cases, of course, that there isn't a pikuach nefesh issue that trumps Shabbat.)
You may or may not agree with the approach, but your question didn't ask for a judgement, only an analysis.
It is possible for a Reform Jew to break Shabbos. From the perspective of traditional Judaism, it doesn't matter whether they believe in the laws. If they don't follow them, they are transgressing.
From a Reform perspective, your description of Reform Judaism is inaccurate. The fact that hilchot Shabbos exist is indisputable. They're written in the Torah, therefore they exist. Reform Jews might deny their authenticity or their obligatory nature, but they cannot deny that they exist. As such, whenever a Reform Jew does something in violation of Shabbos, he is breaking Shabbos, even according to himself. He might not think that it matters though.
The position of the Reform movement is not that halachot like those of Shabbos do not exist. Rather, it is that people should follow the halachot that have meaning to them.