Can a player cast a spell with flash during the beginning of combat step?

Flash (CR 702.8) allows a permanent spell to be cast any time its controller could cast an instant. Instants can be cast whenever a player has priority (CR 116.1, 116.2). During the beginning of combat step, both the active player and the nonactive player receive priority, making it a legal window to cast flash spells.

The beginning of combat step is the first step of the combat phase (CR 507). After the active player receives priority at the start of this step, players may cast instants and spells with flash, activate abilities, and so on before moving to the declare attackers step.

This timing can be strategically important. For example, you might flash in a creature during the beginning of combat step so it exists on the battlefield before attackers are declared — but note that because it didn't begin the turn under your control, it still has summoning sickness and cannot attack (CR 302.6).

Concrete example: Your opponent is about to attack. During the beginning of combat step, you cast Briarbridge Tracker (which has flash) at instant speed. It enters the battlefield, but because it has summoning sickness it cannot attack this turn. However, it is now available to block in the declare blockers step later in combat.

So yes, flash spells are fully legal to cast during the beginning of combat step, as long as a player has priority, which they do during that step (CR 507.1, 116.3b).