Who was allowed to enter the tent of the congregation?

Only the priests were permitted to enter the tent of the congregation. Even Moses, though great in God's service, could not go in once the cloud and the glory of the Lord filled the tabernacle. The presence of God rested there, restricting access to the priests alone.

Les explains that after the whole tabernacle complex was complete and set up, ready for the institution of worship, a cloud covered the tent of the congregation and the glory of the Lord filled the tabernacle. Moses was not able to enter the tent of the congregation. Les emphasizes that it was only going to be left for the priest; even Moses, great man in God's service, could not go in now. The cloud was the very presence of God, the Shekinah glory, which abode thereon and filled the tabernacle. Les reminds that this is the same cloud that appeared when they came out of Goshen and encamped by the Red Sea, protecting Israel while becoming darkness to the Egyptians, and at night it became the pillar of fire. This same cloud now rests over the Holy of Holies, where the Ark of the Covenant is, filling that back room of the tabernacle.

Exodus 40:35

And Moses was not able to enter into the tent of the congregation, because the cloud abode thereon, and the glory of the LORD filled the tabernacle.

Could Moses enter the tent of the congregation after the tabernacle was set up?

No. After the tabernacle was set up, the cloud covered the tent of the congregation, and the glory of the Lord filled the tabernacle. Because that cloud was the very presence of God, Moses was not able to enter the tent of the congregation. That entrance was left for the priest.