Can a marksman use his special abilities when making attacks with a siege engine? Could he, for example, use his finesse style supernatural ability teleporting attack when making an attack with a ballista or catapult?
It depends on the weapon.
Siege Engines come in two flavors: Direct-Fire and Indirect-Fire.
A direct-fire weapon uses a normal attack roll, with the normal penalty for nonproficient use.
Thus, using, say, a ballista, would allow a Marksman to use one to make a Teleporting Attack.
Indirect-Fire siege weapons, however, do not make an attack roll; rather:
To fire an indirect-fire ranged siege engine, the crew leader makes a targeting check against the DC of the siege engine. This check uses his base attack bonus, his Intelligence modifier (if not trained in Knowledge [engineering]) or Knowledge (engineering) skill modifier (if trained in that skill), any penalty for not being proficient in the siege engine, and the appropriate modifiers from Table: Indirect Attack Check Modifiers.
While that check uses the crew leader's Base Attack Bonus, it is clearly not an attack roll (else, the rules wouldn't go out of their way to not use the word "attack"; further, Indirect-Fire siege weapons don't have crit ranges, which reinforces the idea that it's not an attack roll). Catapults are called out as Indirect-Fire siege weapons, so your Marksman could not make a Teleporting Attack with one.