When a game's servers go down, it can usually be attributed to one of two reasons. The first is that the game, for one reason or another, cannot sustain a reasonable playerbase. Another potential reason is that there is some kind of technical issue, which is most often the case. In a very rare turn of events, Hitman: Absolution's Contract Mode servers are going down until further notice because of the passing of legislation and the fact that the servers are not owned by the game's current publisher, Io Interactive.
On May 25th, the General Data Protection Regulation, or GDPR, becomes enforceable in the European Union. Theoretically, the GDPR protects the online privacy of people in the EU, acting as a kind of digital Bill of Rights. Unfortunately, Hitman: Absolution's servers are not going to be GDPR compliant, and given that Io Interactive states cannot find a way to keep the servers running without being in violation of the upcoming legislation, there is no choice but to shut down the servers for the time being, especially given they don't own the servers it was running on. Of course, it brings up the question of why the servers are not GDPR compliant, but it seems unlikely that Square Enix (who owned the servers) was engaging in any particularly nefarious personal data farming operation, but what the situation was on them we don't know - although many games and services are making tweaks to ensure GDPR compliance.
As a result, until Io Interactive announces otherwise, you will not be able to purchase Contracts Mode DLC for Hitman: Absolution in any way whatsoever. While Io wants to make it clear that this recent turn of events "will have no impact on Hitman" (referring to the 2016 episodic game Hitman that is still receiving new content) and that the server closures are due to factors that are outside of their control (like Square Enix no longer owning them and them now having the Hitman IP separate from Square Enix), no firm date has been set for when people can expect things to return to normal. On the bright side, if you absolutely and definitively feel a need to buy the Contracts Mode DLC, you can still do so right now as the servers will be shut down "before the end of May."