Can "nor" be used without "neither"?

You can only use "nor" between two noun phrases if you have already introduced the negative. It is part of the "neither...nor" construct, which is negative form of "either...or". Strictly speaking, too, the form is only permissible when there are two options, but I have seen the count extended by repeating the "or/nor".

Further, don't fall into the double negative: "Neither ...nor.... can" NOT "Neither...nor... cannot...".

So your phrase should look something like "Neither Cummings Motors nor Smith Electric nor our subcontractors can be held liable".

Thinking a bit further, however, this looks like an excerpt from a piece of legal text; in that case there are probably more specialized grammatical structures ("legalese") that you should consider using.