There is one particular style of traditional Japanese comedy called manzai (漫才), which is a type of two-man act. One man is called the boke, who is the buffoon; the jokester; the funny guy. The boke will make jokes, many of which (to American audiences at least) are groan comedy. The other member of the pair is called the tsukkomi, and his job is to react to the boke's jokes (often critically), taking on a role very much like "the straight man" in Western comedic practice.
The usage of tsukkomi in the description you quoted is derived from this. It does not mean that the Vice President is literally the straight man of a comedy duo, but rather that the Vice President is sort of the uptight kind of guy who doesn't laugh at jokes and always shoots down other people's jokes. It's a sort of character archetype that translates only somewhat well into English as "straight man", so you will sometimes see it floating around untranslated.
The tsukkomi is a very common archetype in modern anime and related media (irritatingly common, if you ask me, especially in schlocky light novels). Well-known examples off the top of my head include:
- Kyon from The Melancholy of Haruhi Suzumiya (often playing off of Haruhi and sometimes Koizumi's boke)
- Koyomi from the Monogatari series (playing off against most of the girls at some point or another)
- Basically everybody in Gintama (against everybody else)
- Chiaki from Nodame Cantabile (mostly against Nodame herself)
You can see more about this trope at the TVTropes entry Boke And Tsukkomi Routine.