Oligonucleotide purification with desalting
Oligos from well-known companies are pretty good to my knowledge. Problems begin after 100-300bp oligos/dsDNA fragments.

What I suggest is PCRing your vector like that:

          Primer F: ---------------------__________
vector: ======|v site for 36 bp insert v|==========...
        Upstream                        Downstream
        ______----------------- : primer R

Legend:     
__ portion of primer annealing to vector
-- portion that is your insert of interest


Then you can actually just transform that PCR into e. coli, which will seal two nicks together and propagate your vector. Treating your PCR reaction with Dpn enzyme will ensure that all template DNA is removed.