How does the electron jump across "gaps" in its orbital?
The orbitals, which recently have been observed for the hydrogen atom, are probability distributions. These probability orbital distributions have been calculated using quantum mechanical solutions of the Schrodinger equation which give the wave function, and the square of the wave function is the probability distribution for finding the electron at that (x,y,z,t). This last is a basic postulate of quantum mechanics. Postulates interpret/connect the mathematical model to the physics .
Probability distributions are the same both classically and quantum mechanically. They answer the question " If I throw a dice 100 times how often it will come up six", to "if I measure the electron's (x,y,z,t) how often will this specific value come up". Thus there is no problem of an electron moving around nodes. When not observed there just exists a probability of being in one node or another IF measured.
As others have observed, this goes against our classical intuition which has developed by observations at distances larger than nano meters. At dimensions lower than nanometers where the orbitals have a meaning one is in the quantum mechanical regime and has to develop the corresponding intuition of how elementary particles behave.
The picture of an electron as a little ball that moves around like a billiard ball sometimes works. But it fails enough times that one has to conclude that it's not correct. This is one of those cases when if fails.
The wavefunction represents where the electron might be found if an experiment were done to find it. That's not the same as saying that the electron is actually at some particular place at some particular time. One way of looking at it is to think of the wavefunction as a kind of field that represents the electron itself. The electron field exists anywhere and everywhere there is amplitude. But interactions occur at specific points. If I have some kind of measurement system, it must interact with the electron, and those interaction occur at locations that I might be able to identify.
There are no little balls. There is nothing to cross those lines of zero amplitude.
I don't know how well this picture is justified by theory, but it has to be a step closer to what our theories are telling us than the billiard ball model.
Like garyp says, the electrons are not discrete particles, but rather exist as a smear (a cloud) with the most intensity of their existing in the spaces so described by the wavefunction. Now, all of the electron needs to interact at once, so when an interaction (measurement, chemical reaction, etc.) happens, the wavefunction of the electron changes as well to reflect how it exists at and after the interaction.