Does the federal government allow full and open competition in government contracts?

Yes, that is supposed to be the very purpose of the Federal Acquisition Regulations (the "FAR").

FAR supplements generally require the Government to conduct competitive procurements under "full and open competition" through the use of competitive procedures that are consistent with the need to fulfill the Government's requirements efficiently. See 48 C.F.R. section 6.101. By emphasizing "full and open competition," the relevant statutory provisions and the FAR aim to enable the Government to select the most innovative products and services at the lowest prices.

A statutory and regulatory structure governs contract performance obligations after the contract comes into existence. The regulatory aspects are also primarily governed by the FAR and FAR supplements located in Title 48 of the Code of Federal Regulations