Could discoid galaxies be expanding?
No, for several reasons.


there are no radial velocities in the required range observed, for example in the Milky Way galaxy.
the reservoir of stars moving out from the centre would quickly be drained, so you would need to magically generate them at the centre -- this is much worse than postulating dark matter.
you'd expect huge amounts of stars and gas that have moved out to large radii.


The only serious alternative to dark matter is some modification to Einstein's/Newton's theory of gravity. There are several candidates, but none is really convincing (though arguably, dark matter is not too convincing either).