Is there any word for a person who gives the contract to murder to someone else?

Is there a word for a person who doesn't commit murder himself but gives the contract to murder to someone else?

Since a hitman is sort of a self-employed individual, the person giving the assignment, or contract, could be considered the client.

It seems that the answer to your question is: no, there is not a specific word for that. All the suggestions made so far apply to many more things than just those that hire hitmen.

Consider suborner

someone who pays (or otherwise incites) you to commit a wrongful act

Though, this word was more common in the past. Today, in law, it is mainly used for someone who induces a person to commit perjury.

Technically speaking, suborn doesn't just mean induce someone to conveniently "forget" something in the witness stand, or otherwise get creative with their imagination. An inducement to any kind of crime is suborning, but by far the most common use is in the legal sense above. Or "witness tampering," as the cops call it.

From the book "Searching Shakespeare: Studies in Culture and Authority By Derek Cohen":

Also, employer is commonly used in current vernacular.

From Tvtropes:

But most often, the reason for the Contract on the Hitman is because the employer doesn't want anything linking the killing that the assassin did back to them, and wants the assassin eliminated because — say it with us, people — He Knows Too Much.

From the book "Hitman" By Max Kinnings:

...Firstly, I wanted to introduce myself to you and secondly I wanted to request that you desist from your misguided attempt to collect the bounty that your employer placed upon my head...

From the book "Targeted Violence: A Statistical and Tactical Analysis of Assassinations, Contract Killings, and Kidnappings" by Glenn P. McGovern:

...If a hitman's employer is going to arrange a meeting at a specific time and location...

The person who contracts a murder is a murderer.

At least that is the case in Australia, and I believe, in Britain. The person who instigates the killing is guilty of murder, as is the person who does the deed.

Edit: all of the above assumes that the murder is carried out. Thanks to Frank for his extensive comment :-)

You might want to consider conspirator