Yes, that would, or at least could, be a legal contract.
The key thing that makes a contract between two parties is the agreement, the intention to enter into a binding contract. The written words are only evidence of their agreement, and the specifics of what is agreed to. An oral (spoken) contract can be valid and binding (although in some cases the law requires a written contract).
Electronic contracts do not need to be done through a site such as docusign, although there are advantages to using such a process.
A typed signature will be legally binding if it is intended to represent agreement to the contract. The US federal e-sign law says that no specific technology is required to make a valid electronic signature. See What gives e-signatures legal standing/force in the United States? and https://law.stackexchange.com/a/79670/17500 for more details.
That the contract words are copied will also not impair the validity of a contract. Many bushiness use form contracts. Many lawyers create new contracts by putting together parts of old contracts that served their purpose.
The person sending the offer and proposed contract must make sure that the other party understands that this is to be a binding contract, and agrees to the use of an electronic signature, and agrees to the contract as a whole.
It would be a good idea if a bit more detail were included than in your example. Must the agency find people acceptable to the client? Haw soon must it find them? How much must the client pay? How soon must it pay? A good contract will specify such details.