Can a body which has no initial velocity move just with acceleration?
Not sure what you mean. Once it starts moving it will have both acceleration and velocity.
You can certainly have a body with zero initial velocity begin moving under an acceleration. Drop a ball.
"Acceleration" means "change in velocity. As long as it has acceleration, even if it has no initial velocity it will immediately have non-zero velocity.
A motionless freely-floating body can also spin without moving when given only angular acceleration (in zero gravity). See
[added] Angular acceleration requires a pair of forces (a couple).
Separate names with a comma.