How much does the electric current within the Earth' core depend on the axis of the Earth's orbit?
There's still a lot that isn't understood about the Earth's magnetic field, so there's some uncertainty to any answer to your question.   That said, I touch on some of it.   


  isn't the current that causes the magnetic field affected by the
  Earth's axis


I'd say probably not.  We can see this by looking at other objects with magnetic fields relative to their axis.  There are some theories that planets need to rotate to have magnetic fields, but Mercury rotates very slowly and it has a small magnetic field.  Mars rotates with a similar period to Earth and it lost it's magnetic field about 4 billion years ago.



and Neutron Stars.


  Often, the magnetic field is not aligned with the spin axis


The Magnetic Axis and the Axis of Rotation seem to have very little to do with one another.  The Rotation Axis' relation to the plane of the solar system, I suspect that has even less influence, but I have no idea how to go about explaining the why behind that, it just feels improbable to me because the effect seems more internally generated and the inside of the Earth cares which way it spins and stretches, but not how it lines up with the plane of the solar system - (I would guess).

In addition, the Earth's Magnetic North Pole moves around.  Earth's Axis changes very slowly.  It's difficult to see a relation between the two beyond "in the same general area".




  (or perhaps the orbit around the sun)?


It's always good to explore possibilities.  Pretty interesting, short article on Mars' loss of it's Magnetic field here that suggests it's the inner workings of the planets that drive magnetic fields.

When there's enough temperature variation and a liquid medium, spirals (or swirles) can form because circulation is more efficient than convection at heat transfer.  Like the atmosphere, you also have the hotter material wanting to expand and rise and the cooler more condensed material wanting to fall, but you also have increasing pressure as you go down, and, perhaps, not uniform molecular density, so it's a pretty tricky subject requiring lots of math behind when spirals form and when they don't.  The solidifying of the other core into the inner core and heat released from that phase change may play a role too.  (I read that, will post a source if I can find one).



Earth's relatively rapid rotation (and corresponding Coriolis effect) might keep the magnetic field pointing roughly North-South (I would guess) but as noted in the map above, there's a fair bit of wobbling and in addition to that, every few hundred thousand years the field flips.  

The Moon may cause the movement of the magnetic poles too.




  (or perhaps the orbit around the sun)?


(yes, I just quoted you twice)

That article just above suggests the stretching and squashing of the Earth, both due to the Moon and the Sun would have an effect, so in a sense, the Orbit around the Sun might affect the magnetic field, but more the tidal effects from the Sun than the actual orbit itself.

Because this field remains unresolved, we should never say never until it's better understood.