How can something be an acid if it doesn't have a hydrogen to donate?

The Ni^2+ ion is an electron-deficient species. Under Lewis acid base theory, it is an acid. Other Lewis acids include Al^3+ (as AlCl3 usually), Fe^3+ (usually as Fe(NO3)3). BF3 is another Lewis acid.

Lewis bases are electron-rich species. Examples include NH3 and H2O. Interestingly, double bonds (being an electron-rich area) can function in a Lewis base way in certain reactions.

You must make it clear what type of acid you are referring to:

Brønsted acids are H^+ donors; Lewis acids are e⁻ pair acceptors.

NiCl2 as a Brønsted acid:

NiCl2(s) + ∞H2O → [Ni(H2O)6]^2+ + 2Cl^-(aq)

[Ni(H2O)6]^2+ + H2O ⇋ [Ni(H2O)5(OH)]^+ + H3O^+ pKa = ~10 a weak Brønsted acid.

As a Lewis acid: H2O:→Ni^2+ as in [Ni(H2O)6]^2+; H3N::→Ni^2+ as in [Ni(NH3)6]^2+

Cl:→Ni^2+ as in [NiCl4]^2-