Whether an exodus happened is a separate question from who wrote about it, and the evidence is thin in a specific and instructive way.
No Egyptian record mentions Israelite slaves, a departure, plagues, or a drowned army, though Egyptian royal inscriptions do not generally record defeats, so silence carries limited weight. The Merneptah stele of about 1207 BCE names Israel as a people in Canaan, which is the earliest external mention of the name and puts an entity called Israel in the land by that date. Archaeological survey of the Sinai has found no trace of a large population moving through it or camping for a generation, and the sites the narrative names as occupied at the time of the conquest were, in several cases, unoccupied then.
The account of Israelite origins that now commands the widest support is that the early Israelites emerged largely from within Canaan, from highland settlements that appear in the archaeological record from around 1200 BCE and are distinguishable from lowland Canaanite sites mainly by the absence of pig bones and by pottery of local descent. On that model the exodus narrative is a foundation story rather than a report.
A minority of scholars argue that a smaller group did come out of Egypt and that its story was adopted by the wider population. Points in its favour: Moses, Hophni, Phinehas and Merari are Egyptian names, not Hebrew ones, and no one invents an origin story about having been slaves.
The figure and the event are contested. Saying so is not the same as saying either is disproved.