[1] "I don't know what is the best vehicle for that."
[2] "I don't know what would be the best vehicle for that."
[3] "I don't know what be the best vehicle for that."
My ear says [2] would be best. But [3] is shorter; is it also acceptable?
[1] "I don't know what is the best vehicle for that."
[2] "I don't know what would be the best vehicle for that."
[3] "I don't know what be the best vehicle for that."
My ear says [2] would be best. But [3] is shorter; is it also acceptable?
There are a number of ways to express the idea contained in your example sentences without sounding unnatural:
"I don't know what the best vehicle for that [thing] is."
"I don't know what the best vehicle for that [thing] would be."
"I don't know what would be the best vehicle for that [thing]."
"I don't know the best vehicle for that [thing]."
"I can't figure out the best vehicle for that [thing].
among others. Your option 1—
"I don't know what is the best vehicle for that [thing]."
—sounds a little awkward to me, but there is nothing syntactically wrong with it. As mplungjan remarks above, "I don't know what be the best vehicle for that [thing]" has a Long John Silver ring to it and would not be a very suitable way to express the idea in modern idiomatic English.