Depends on how convenient your vector is to PCR, but the way I would do it personally is just to amplify the entire vector using a reverse primer that binds to the 3' end of the first gene and a forward primer that binds to the 5' end of the second gene. At the 5' end of each primer I would have half of the coding sequence for the linker as an overhanging sequence. One of the primers would be phosphorylated (either at the time of synthesis by the company or prior to PCR using T4 PNK), and I would just blunt end ligate it and transform it.

This should give you a rough idea of how to do it:

https://www.protocols.io/view/around-the-horn-pcr-and-cloning-261geqx7g479/v1?step=11

Though I would rather do any phosphorylating prior to the PCR rather than after, as T4 PNK works much better on ssDNA than dsDNA without overhangs. The extension time given is rather excessive for Q5 which should need at most 20 seconds/kilobase of length. Another caveat is you only need to do DpnI digestion if you are not going to gel extract the product. Gel purification alone is enough to remove the template plasmid.

This general method of cloning is pretty adaptable to a lot of different conditions so once you understand it, I wouldn't hesitate to modify it to whatever way is the most convenient for you.

Dear Subeksha Subedi

i'm agree with Alexandra Johnson

if you vector size is not too big (less than 8Kb) you can perform a vector PCR adding in the primers 2 overlapping flanking regions that contain the sequence of the GSGS link that you would like to add using an enzime free cloning approach named PIPE cloning

I used this approach in the past to add N-terminal signal peptide to protein secretion to genes or other TAGS (e.g 10X His) or linkers and it worked quite well. Of course since you have a repetition of a lot GS, to reduce the probability of wrong coupling of the 2 overlapping regions you have to desing a DNA sequence using different codons for the 2 aminoacid to make the sequence less repetitive.

Yoo can find more information about this approach and the PIPE cloning in the following links:

https://proteocool.blogspot.com/2021/09/tips-of-week12-rapid.html

https://www.blogger.com/video.g?token=AD6v5dymTB4mmANTKeTOyzXQq7QSUurTLQFPDsGxi4FqyNDpu3YZutcuN0VfLTfDTPRD8edEIDPD6bJZIe_VWirwCNZHkJ1A34BATcDIzXZEbV1UUtRLR6Nc19tU2_t6-Or5Sag2dwU

or in the following articles:

https://pubmed.ncbi.nlm.nih.gov/18988020/

https://pubmed.ncbi.nlm.nih.gov/18004753/

good luck

Manuele