For a Design Flow Chart to cover the ASME Code it would be rather extensive. Just address the components of the vessel in accordance with the Code and then the overall vessel in accordance with the Code.
Buy the PTB standards which cover example calculations to ASME VIII Div 1.
Appendix L contains some sort of flowchart for joint efficiencies.
Google around for books on pressure vessel design and buy them.
Compile your own flowchart.