What is a single word for a person who makes contracts?
It is not exactly clear why you need a single word, so it is a bit hard to know what word would fit best.
Part of the problem is that "a person who makes contracts" is ambiguous.
On one hand, a person who secures an agreement to basic terms of a contract may be called a salesman, broker, negotiator, deal-maker or even at the low end of the contract world a clerk or a sales clerk, since the most common kind of contract is a sale. Of those, broker or deal-maker are the most general.
But, someone who causes an agreement that is reduced to writing in a written contract is often not the person who does the writing. Sometimes, the reduction to writing is done by some sort of legal professional who has input into what the secondary terms of the deal in the contract should include and how they should be written, and sometimes by someone more along the lines of a secretary who strictly follows the deal-maker's directions.
It is not clear what part of making a contract is intended.
In legal opinions, a person who writes a contract is often called a drafter, or if the person who is bound by a contract writes it him or herself, a party to the contract. Both of those terms refer to a "role" that a person takes in connection with a contract.
Sometimes the word party or drafter is used in a sense which deliberately ignores the distinction between the person who physically writes the document and the person who is bound by it, treating them instead as a single unit.
A scrivener or scribe is a person who literally writes down the words of a contract or other document, at the direction of another (usually the person who is in substance the preparer of the document) either from dictation or at any rate with minimal professional discretion in how this is done. Either of these descriptions can apply to someone who writes letter or other documents to dictation or with little discretion, however, and not just to people who draft nothing but contracts.
In governmental bureaucracies, the job description of someone whose duties consist of writing contracts without being a lawyer, and nothing else, is usually a contracting technician or a legal technician, or a paralegal.
A lawyer, regardless of practice environment in the U.S., who predominantly writes contracts is called a transactional lawyer (a contract lawyer is a lawyer who works as an independent contractor for a law firm as opposed to an attorney on a job by job basis, not necessarily a lawyer who drafts contracts).
Similarly, a contractor is someone who is in the business of performing work on a contract basis and does make that deal with the hiring party, but often a contractor will spend little if any time drafting the contract which may instead be drafted by the hiring party in many circumstances.
There is no parallel term to litigator for a transactional lawyer.
In most countries of the world whose legal system is not derived from England, there is a separate profession called a notary which in those countries is not the mere signature authenticator that such a person is in the United States, but is a trained legal professional who does transactional legal work and also serves in a capacity analogous to the secretary of state of a state government or a clerk and recorder of a county government as an official record keeper for public documents. In those countries, an attorney or advocate is mostly a litigation attorney.
What is a single word for the study of making contracts?
I do not believe that there is a single word in common use with that meaning. People who do so are usually law professors or business economists or business law professors (outside law schools, often in a management department of a business school).
For example, I have not seen such a term used at the ContractProfs Blog which is run by law professors whose teaching and research specialization is in contracts, a place one would surely expect such a word to be used widely if it existed.
There is a term of art called Private Law which is used to refer to the area of law that governs the relationships of individuals and entities who are not government entities, and many people who study contracts would call themselves "Private Law scholars", and there are many people who study contracts who describe themselves as economists, but neither of those terms are specific only to the study of contracts.
It also isn't clear if you mean the study of contracts in a way that would include contract law (usually done by law professors) or in the study of what they say empirically (usually done by economists or business professors).
Few people study contracts globally and in all respects as their primary research field. Often, people study only particular kinds of contracts. And, people usually study contracts either from the perspective of contract law as law professors or undergraduate business law professors, or from an economists perspective, not both. While a paleontologist is a person who studies fossils in the field of paleontology, and a person who does stonework is a mason engaged in the field of masonry, there is no analogous construction for people who write contracts or study them.