I'm trying to state in one sentence several things that are lacking.
There's no A, or B, or C.
There's no A, no B, and no C.
Are these both grammatically correct? What's the difference?
In general use, those expressions are both pretty normal and pretty much equivalent, regardless of what any logician makes of them. They seem grammatically correct to me assuming appropriate phrases are supplied for A, B and C.
From a logical perspective, given that A' (not A) is true, and B' is true, and C' is true, the disjunction and the conjunction give the same answer (true), so in fact the sentences are equivalent if the (negated) assertions are correct. If one of the premises is incorrect, then the overall result differs depending on whether AND or OR is used as the connective.
There's no A, B or C.
The "no" applies to all items in the following list.
For lunch I have a plain burger. There's no mayo, mustard or ketchup.