Can you use "nor" if the sentence does not have a "neither" but instead some other negative word?

According to The Cambridge Grammar of the English Language p1309,

Nor appears as a coordinator paired correlatively with neither, or non-correlatively as a variant of or in negative contexts.

As this is clearly a negative context, the use of nor is prefectly acceptable.

Examples of this usage may be found in quality writing:

Ultimately, I saw a system that often does not have the time nor the inclination to care about the kids inside it. (Teaching doesn't count; Sachar, E;Washington Monthly;1990 (Sep))

The reason the CPA firm was willing to let the clients go was that the firm at the time did not have the capability nor the desire to serve small- to mid-sized clients that needed help in software selection, set up, maintenance, and training. (Growing a Nontraditional Accounting Firm: Warren and Associates; Bagranoff, Nancy; Turner, Leslie; Journal of Information Systems; 2004 (Fall))

People who suffer from this kind of political and social upheaval do not have the time nor the enthusiasm to admire their artists. ( Mario Bencastro on the character of words; Mujica, B; Rascon, S.G.;Americas; Vol. 43 Issue 4, p24; 1991)