This largely depends on what you mean by ‘logically prove that anything exists’ (and a bit on what you mean by ‘logical object’). Let’s consider two alternatives.
If you mean: ‘showing that it is a logical truth that something exists’, then the answer is no, or almost no. If a is an individual constant of given logical language, a = a is a logical truth, and so is the existential sentence ∃x x = a – which effectively say that a exists. So, it is a logical truth that something (or, some thing) exists; but that’s as far as it goes: we can’t prove the existence of a second thing, nor can we prove that the existing thing is a physical object.
Some logicians feel that even this is too much. Logic should be subject neutral, they argue, to the extent that the existence of no thing should be a logical truth. This type of approach is known as Free Logic.
On the other hand, Logicists and Neo-Logicists hold that numbers exist of logical necessity, and have tried to show that their existence follows from what they take to be laws of logic. If this can be done successfully, numbers would be 'logical objects' - but that certainly doesn't mean that their existence is trivial! This is one of the best-developed, but also one of most complex and complicated views in formal logic and philosophy of mathematics.
Alternatively, by ‘logically prove that anything exists’, you might mean: ‘prove deductively as opposed to inductively that something exists’. The next question is then what you want to prove this from, i.e.: What are the premises in your proof? If they are once again only logical truths, we are back to the above situation. However, logical truths aren’t the only truths. Take the claim ‘Proxima Centauri is a red dwarf’ as an example. This is certainly not a logical truth, but a truth nevertheless. Now, both the existence of Proxima Centauri itself, and the existence of red dwarfs, follow from this statement. Importantly, they follow by means of a logical rule of inference, or deductive rule, known as existential generalisation.
Just because we’ve used a logical rule in the proof doesn’t mean that the conclusion are truths of logic: that Proxima Centauri and red dwarfs exist, are contingent truths and not logical ones. So, we can prove deductively that something exists, even though the existence of that something isn’t itself a truth of logic. (But that's all we wanted.)
Of course, we haven’t proved the existence of something that we previously thought didn’t exist. However, Quine in his famous paper On What There Is points out that the same method will sometimes uncover ontological commitments that we previously didn’t think we had to make. In particular, he argues (roughly) that because the laws of physics involve numerical statements (like e.g. ‘The speed of light is 299.792.458 m / s’), physics commits us to the existence of numbers. This is called the Indispensability Argument.
You mention God, and proofs of the existence of God; but as it stands, I’m not sure you’re your question is specific to God’s existence. Also, people have certainly done more (whether successfully or unsuccessfully) than tell stories about God: proofs of the existence of God abound, and some even claim to be ‘logical’ or at least ‘analytic’ or ‘a priori’. – Whether these proofs are sound, is a different question, of course.