According to the US Tennis Association Rules of Tennis, "Immediately before starting the service motion, the server shall stand at rest with both feet behind (i.e. further from the net than) the baseline and within the imaginary extensions of the centre mark and the sideline." Under the categoy of "Foot Fault" the rule is as follows:
During the service motion, the server shall not:
Change position by walking or running, although slight movements of the feet are permitted; or
Touch the baseline or the court with either foot; or
Touch the area outside the imaginary extension of the sideline with either foot; or
Touch the imaginary extension of the centre mark with either foot.
If the server breaks this rule it is a "Foot Fault".
The server is allowed to have one or both feet off of the ground during a serve. However, in doing so, they place themselves at risk of being called for a foot fault. The official comment on this rule is:
USTA Comment 18.2: What does the rule mean when it says that the server may "not change position by walking or running"? One key to understanding this rule is to realize that the server's feet must be at rest immediately before beginning to serve. The delivery of the service then begins with any arm or racket motion and ends when the racket contacts the ball (or misses the ball in attempt to strike it). To define walking or running with precision is difficult. This rule is intended to prevent the server from taking advantage of the receiver by serving while "on the move" and requiring the receiver to guess the position from which the serve will be launched, and the rule should be enforced with that intent in mind.
A server who takes more than one step with either foot after the "feet at rest" position described above is at risk for being called for a foot fault. The serve becomes a foot fault when, in the judgment of an experienced official, the server has materially changed position before or during any racket or arm motion.
A server whose footwork changes significantly from one serve to the next is at risk for being called for a foot fault.
Serves that look like the running volleyball serve violate the rule. Serves in which the server runs or walks from a point well behind the baseline to the baseline are also illegal, as are serves in which the server walks or runs along the baseline before choosing a spot from which to deliver the serve.
So, technically, while leaving the ground with both feet is legal, a jump serve would be considered illegal.