I'm not happy with this sentence for several reasons, one being that there are not enough commas to separate the clauses. I can decipher, for instance, that the main clause is the first part of the sentence: It will not deal or negotiate with any similar competitive businesses or project. It has a subject and a predicate. Note that left standing on its own, it is grammatically correct. The rest is what would be called a subordinate clause: directly or indirectly, in any capacity, or for any other person, in connection with.
Perhaps the sentence should read as: It will not deal or negotiate with, directly or indirectly, in any capacity, or for any other person in connection with, any similar competitive business or project.
In answer to your question, the object in this sentence would be 'any similar competitive business or project'.