Why is Aluminium Chloride a Lewis Acid if it's an ionic compound?
if it's an ionic compound


Actually, it is debatable. $\ce{Al^{3+}}$ is a very small cation and its formation is strongly unfavored. In fact, this cation, just like many others, may exist only in solvates, like $\ce{[Al(OH2)6]^{3+}}$, or unchemical environment. 

In particular case of anhydrous $\ce{AlCl3}$ the molecule is a strong Lewis acid, strong enough to coordinate with lone pair of another similar molecule, so it forms either dimer $\ce{Al2Cl6}$ with two chlorine bridges, donating one pair to both Al ions, or polymer, with Al atoms exhibiting octahedral coordination. Unlike many ionic compounds, melted $\ce{AlCl3}$ has poor conductivity. 

This, however, does no apply to $\ce{AlCl3.6H2O}$ which is, again, a ionic compound $\ce{[Al(OH2)6]^{3+} (Cl^{-})3}$ and does not exhibit any Lewis acidity (though it does exhibit strong Brønsted acidity, i.e. tendency for hydrolysis)

This said, however, I have to clarify: ionicity of compound does not prevent particular ion from being Lewis acid/base. In fact, cations often are excellent Lewis acids. The only thing needed for a particle to be a Lewis acid is presence of low  vacant orbital, and sometimes presence of such orbital is not derivable from octet rule. For example, $\ce{SnCl4}$ is Lewis acid, though in it $\ce{Sn}$ has full octet. The orbital in question is describable in terms of MO theory which I strongly encourage to study. 

The most often considered Lewis acid is hydrogen cation. In fact, this Lewis acid is so strong, that it never occurs free, it is always bound to Lewis base (again, except unchemical environment). Another common Lewis acid is $\ce{Na^+}$. This cation is quite weak Lewis acid, but it is sterically accessible, and usually is paired with anion, which is usually a Lewis base. This said, despite mostly ionic nature of $\ce{NaCl}$, there is considerable covalent bonding in it. (of course, on the other hand there is such anion as $\ce{BPh4^-}$ which is not a lewis base of considerable strength, so its salts are almost purely ionic in nature)