Theoretically, yes.
As /u/tiredstairs points out, Article II of the UN Charter calls on countries to:
"...refrain in their international relations from the threat or use of force against the territorial integrity or political independence of any state..."
unless they're acting in self defense (article 51).
Chapter VII of the UN Charter gives the Security Council the ability to call for war against a particular country. If it does so, an aggressive war against that country would be considered a "legal" war.
The Security Council has only used this ability twice, once in 1950 (Resolution 82 / the Korean War), and again in 1990 (Resolution 678 / the First Gulf War).
However, if a country were to wage an aggressive war without the Security Council's permission, there's no mechanism to automatically levy sanctions, deploy peacekeepers or take any other action against the aggressor country.
In order to impose any kind of consequence, the UN Security Council would need to pass a resolution authorizing such action. This is often easier said than done, since the 5 Permanent members of the Security Council (the United States, Great Britain, France, Russia, and China) have veto rights. So, even if a vote was 14 to 1 in favor of a resolution, it wouldn't pass if just one of the Permanent members disliked it.
Therefore, if a country has the support of a majority of the countries on the security council, or the support of one of the Permanent members, (or if it is a permanent member itself), then it can essentially wage an aggressive war without any practical consequences (from the UN at least) even though it's technically an "illegal" war.
Alternatively, even if a country doesn't have any support in the Security Council, it could still decide to go to war and just deal with the consequences. Perhaps its economy is already isolated to the point that sanctions won't have much effect; or perhaps it's located in an obscure part of the world where countries would be reluctant to send troops as part of a UN peacekeeping mission.
No, Member States are not required to request permission from the Security Council or any other U.N. organ in order to declare war on another. However, according to Article 39 of the U.N. Charter the Security Council is responsible for determining what actions should be taken in order to "maintain or restore international peace and security" if a "threat to peace" is determined to exist. During the Cold War the Security Council has acted on Article 39 by condemning the actions by North Korea leading to the Korean War in S/RES/82 - S/RES/85.
http://daccess-dds-ny.un.org/doc/RESOLUTION/GEN/NR0/739/16/IMG/NR073916.pdf?OpenElement - General Assembly Resolution 3314 (Definition of Aggression)
http://www.un.org/en/sc/documents/resolutions/1950.shtml - Complaint of aggression upon the Republic of Korea
Edit: Wrong Link