Do Jews believe that God appeared to Moses?
Keep in mind the illustration of The Miracle Worker. The Bible is the “play” version of the true story. It tells us things that are true, but it often employs narrative devices to get these points across.
Something did happen historically speaking. But is what we read in Scripture about Moses at the burning bush, is this some literal description? Even with those Jews who read the Exodus story more down the literalist path don’t read it at face value (for instance, literalist readings in Judaism see a contradiction in the Exodus text between 3:2 and 4, where one verse says it was an angel at the burning bush and the other says it was God).
What is commonly accepted is that Moses had some type of revelation, whether corporeal (visible to bystanders) or a vision (it is generally held Moses was seeing not a “fire” at the burning bush but the Shekinah) or that even the text is dramatizing the enlightenment Moses received upon learning that Jethro was the high priest of the same God worshipped by Moses’ ancestors, this is what the text is saying. Moses had a “revelation.” The actual nature of this will likely remain unknown, but it was truly a theophany and could have included aspects of some or all the above. (A “theophany” is a manifestation of the Divine as opposed to an “epiphany” where one comes to realize that the Divine is hidden in the guise of something else.)
Do Jews believe that God appeared to anyone among them instructing them to start their religion?
The answer to that is no. While it is part of our history that certain individuals experienced theophanies, there is no instruction for any "let's start the true religion" or anything like that.
This is not Mormonism. That religion is based on someone who claims that God instructed their leader to start that movement. But not Judaism. God didn't ask anyone of us to start the religion of the Jews, not Moses, not Abraham, no one. Not specifically anyway.
Part of that has to do with our concept of God. In short, since Abraham onward Jews have understood God to be the opposite of all deities worshipped by the nations. The physical universe is an effect caused by "something," and that Great Cause is what God is to us. While we refer to this Cause as "God," our concept of God is more of the "un-God," like the soft drink 7-Up used to be called the "un-cola." Nothing humans think a god is or can be describes the true God. God is greater, far more incomprehensible, yet closer and relatable than any god can be. In a sense there are no such things as gods. But the Great Cause of all we see, that is God.
This explains why so many things in Jewish worship seem similar to other religions but then have a facet or do something the complete opposite way others do. For instance, our God has a name, but we don't pronounce the name. Our God is greater than all others, but has no image. Our God is a mystery, but we can still relate to God.
The Jewish response to the God concept is what has become our religion. And it wasn't a religion to begin with. Like other nations of the ancient world, our God was connected to our national identity and ethnic customs. The worship of the Hebrews technically didn’t become what we know as a “religion” until after the Babylonian exile. Granted an entirely new thread can be developed to debate the ins and outs of that, but before then what the Jews did in worship of their God was linked to their ethnic or cultural identity. Jews worshipped their God, and people of other nations worshipped theirs. Rarely did a person from a different nationality worship the deity of another nation. Things didn’t work that way as they do now. The worship of the Jews was no different in the religious paradigm of the ancient world. In fact, it was very much limited by it, shaped by it, and governed by it.
While this doesn't mean Jews don't hold that God did not revealed things regarding our religion. God did indeed do this, sometimes more directly than others, but our response to God began the moment we became aware of this Great Cause. Whether it was a formal act of liturgy or the mere refusal to worship the images of heathens, the worship of the Jews began when they became aware that there was something greater than all the religious ideas humans could come up with.