Can a player in a Commander game use Doomsday and then win with a Laboratory Maniac in the same turn?

Doomsday (from Weatherlight) lets you search your library, exile it, then stack exactly five cards from among any cards in your graveyard and library as your new library. This happens as part of resolving the spell. Critically, after Doomsday resolves, Laboratory Maniac is not yet on the battlefield unless you put it into play some other way.

Laboratory Maniac's replacement effect (CR 614.1) says: if you would draw a card while your library is empty, you win the game instead. For this to work, Laboratory Maniac must be on the battlefield at the moment you attempt to draw from an empty library (CR 112.1, 614.1a). If it is sitting in your Doomsday pile, it cannot apply its effect.

The classic 'Doomsday pile' solution is to build the five-card pile so that cards within it (like Gitaxian Probe, Dark Ritual, Lotus Petal, Laboratory Maniac, and a draw spell) let you cast and resolve Laboratory Maniac first, then draw a card with an empty library. In Commander, this also requires the commander's color identity to allow black (for Doomsday) and blue (for Lab Man), or you use a commander that enables this.

Concrete example: You cast Doomsday, stacking your pile as [Lotus Petal, Dark Ritual, Laboratory Maniac, Gitaxian Probe, Island]. You then cast Lotus Petal, crack it for U, cast Gitaxian Probe paying 2 life to draw Lotus Petal... actually, you play through the pile, eventually getting Laboratory Maniac onto the battlefield, then cast Gitaxian Probe (or another draw effect) with your library empty — Laboratory Maniac's effect triggers and you win.

In Commander specifically, note that Doomsday puts you at very low life (it costs half your life, rounded up, per its Oracle text), so you must ensure you survive to execute the combo. No Commander-specific rules alter how Doomsday or Laboratory Maniac function (CR 903), so the interaction works the same as in other formats.