According to Wikipedia Blankenship was nominated virtually in 2020, however, the Constitution parties of Virginia and New Mexico instead gave their presidential nominations to Sheila Tittle and the Virginia Constitution Party gave its vice-presidential nomination to Matthew Hehl. The South Carolina Constitution Party chose to not run a presidential candidate during the 2020 presidential election.
With that being conveyed and no main website available, I'd say that the party has fallen by the wayside and perhaps still remains only in some states.
I have contacted the FL Constitution Party to see if I can get an answer to what has happened.