The MIT synthetic chemist Gobind Khorana won the 1968 Nobel Prize in Chemistry for his work which successfully was able to make chains of Ribonucleic acids. The chemistry was difficult at the time but he won the prize for making specific sequences of RNA bases which were then fed to cells, resulting in specific amino acid chains, which ultimately deciphered the specific base to amino acid correspondence of the genetic code.
Khorana was only one of the chemists who worked on nucelotide synthesis, but the technique overall is simple and since nucleic acid polymers are only unbranched polymers,
any sequence combination is easily obtained. Each sequential reaction adds one base. Repeat the reaction as many times as needed.
The chemistry Khorana devised was refined and easily extended to DNA polymers since the linkages are identical, the difference between DNA and RNA is only a single alcohol (-OH) group, which is not directly involved in the bond between the nucleotide bases.
By 1983 when Kary Mullis concieved the idea of PCR, short sequences of DNA and RNA could be synthesized on commercially available machines. In fact Mullis was responsible for just such a machine at Cetus Pharmaceuticals and had been trying to find a way to increase demand for his services.
The PCR oligos were the easy part of the operation. To prove the concept, the team at Cetus used a conventional DNA polymerase and add it to the PCR mix after every cycle. By 1986 the Cetus team had succeeded in performing PCR a heat tolerant DNA polymerase from Thermus aquaticus, a bacterium from a hot spring, which basically made it the technique we still use today.