How did Gollum enter Moria?

Question

We know from the books that Gollum already was in Moria when the Fellowship entered. We also know that he did not enter from the West but from the East.

However, as far as I know, we do not know exactly how he entered. For instance, that he managed to sneak past the guards above the stairs and somehow made his way across the bridge. So, how did he do it?

I'm looking for a canon answer.

Answer 1

This is all we know.

It seems clear that pursued both by Elves and Orcs Gollum crossed the Anduin, probably by swimming, and so eluded the hunt of Sauron; but being still hunted by Elves, and not yet daring to pass near Lŏrien (only the lure of the Ring itself made him dare to do this afterwards), he hid himself in Moria.10 That was probably in the autumn of the year; after which all trace of him was lost.

What then happened to Gollum cannot of course be known for certain. He was peculiarly fitted to survive in such straits, though at cost of great misery; but he was in great peril of discovery by the servants of Sauron that lurked in Moria,11 especially since such bare necessity of food as he must have he could only get by thieving dangerously. No doubt he had intended to use Moria simply as a secret passage westward, his purpose being to find ‘Shire’ himself as quickly as he could; but he became lost, and it was a very long time before he found his way about. It thus seems probable that he had not long made his way towards the West-gate when the Nine Walkers arrived. He knew nothing, of course, about the action of the doors. To him they would seem huge and immovable; and though they had no lock or bar and opened outwards to a thrust, he did not discover that. In any case he was now far away from any source of food, for the Orcs were mostly in the East-end of Moria, and was become weak and desperate, so that even if he had known all about the doors he still could not have thrust them open.12 It was thus a piece of singular good fortune for Gollum that the Nine Walkers arrived when they did.
_{(Unfinished Tales, The Hunt for the Ring)}

The East Gate doesn't seem to have been hard to enter. Both Gandalf and Aragorn had previously entered there without trouble. It's not clear that the Orcs did keep a perpetual watch and a guard on the gate.

Yet it will not be the first time that I [Gandalf] have been to Moria. I sought there long for Thráin son of Thrór after he was lost. I passed through, and I came out again alive!
_{(The Lord of the Rings, A Journey in the Dark)}

and

‘I too once passed the Dimril Gate.’ said Aragorn quietly; ‘but though I also came out again, the memory is very evil. I do not wish to enter Moria a second time.’
_{(The Lord of the Rings, A Journey in the Dark)}

At any rate, Gollum was a practiced and expert sneak-thief, who had spent many lifetimes evading Orcs.

Answer 2

He would have found an unguarded entrance

There is no reason to believe that Gollum entered Moria by the main east entrance (over the bridge that the Fellowship uses to leave Moria). As mentioned in the question, that would be very difficult, even for the stealthy Gollum, if the orcs kept any sort of watch on it.

Systems of caves and tunnels as large as Moria are unlikely to have one outlet in the east and one in the west. They would typically have several outlets of various sizes. Even if that was not the case, the dwarves would probably have created secret entrances (leading to narrow, easily defended passages) that they could use in emergencies (similar to the postern-door created by the Rohirrim in the Hornburg).

We know that Gollum was adept at finding his way into such places. We are told how he found his way into the caves where he eventually met Bilbo.

‘But as he lowered his eyes, he saw far ahead the tops of the Misty Mountains, out of which the stream came. And he thought suddenly: “It would be cool and shady under those mountains. The Sun could not watch me there. The roots of those mountains must be roots indeed; there must be great secrets buried there which have not been discovered since the beginning.”

‘So he journeyed by night up into the highlands, and he found a little cave out of which the dark stream ran; and he wormed his way like a maggot into the heart of the hills, and vanished out of all knowledge. The Ring went into the shadows with him, and even the maker, when his power had begun to grow again, could learn nothing of it.’

_{The Lord of the Rings Book One, Chapter 2: The Shadow of the Past
Page 54 (Houghton Mifflin Harcourt; Single Volume 50th Anniversary Edition)}

Conclusion

Gollum would have entered Moria by finding a small unguarded entrance and then "worming his way in"