Is it possible to ligate a vector cut by 2 different blunt ended restriction enzymes?

Why shouldn't it be possible?

Will it be of good efficiency????

It will be of lower efficiency than sticky end.

So, I already have a promoter cloned in a vector and I intend to study expression patterns by doing promoter bashing where I digest the DNA with blunt ended enzymes and removing the small piece and re ligating the remaining fragment.

ligation shouldnt be an issue if your are just excising a fragment, as long as you do the obvious, i.e. gel purification of the vector, otherwise you could just ligate the excised fragment back into it's original position.