Is it possible to use a semi colon within a list?

It is acceptable to use semi-colons in a list. However, the semi-colon is a stronger separator than a comma, so your example is effectively making a list which separates the items like this (which doesn't make sense):

gene1
kidney disease, gene2
cardiac arrhythmia, gene3
arterial stiffness, gene4
aortic aneurysm

It would be better to swap the commas and semi-colons:

'These genes are significant in related conditions (gene1, kidney disease; gene2, cardiac arrhythmia; gene3, arterial stiffness; gene4, aortic aneurysm) suggesting new avenues of work.'

Alternatively you could replace the commas with dashes:

'These genes are significant in related conditions (gene1 - kidney disease; gene2 - cardiac arrhythmia; gene3 - arterial stiffness; gene4 - aortic aneurysm) suggesting new avenues of work.'