The sentence is both grammatical and logical.
The speaker is evaluating something, buying cookies (which is a gerund phrase used as a noun). She is determining that the act of buying cookies will not accomplish something. It has the same logic as
He is hungry. Eating will solve that.
His eating will solve that.
This is functionally the same as
He will solve that by eating.
Obviously there is someone (or something) doing the paying. But that is always the case with gerunds. The person or thing doing or experiencing the gerund is often left unstated, and perhaps even unknown.
The original sentence could be expanded to
I don't believe that [our] buying cookies will deliver on our promise.
The gerund is the subject of the clause that X will deliver on our promise.
Yes, the first sentence is equivalent to
I don't believe that by/with buying cookies we will deliver on our promise.
The difference is structural, not logical. The gerund phrase, buying cookies is now part of an adverbial phrase by buying cookies that modifies will deliver rather than being the subject of will deliver in the first sentence.