The short answer
Light rain can be workable if the BBQ is stable, ventilated, and away from anything flammable, and if you can keep food covered. Heavy rain, thunder, strong gusts, or water pooling near the grill are all good reasons to wait.
Safe shelter matters
Never BBQ indoors, in a garage, in a tent, or under a low enclosed cover. Carbon monoxide is invisible and dangerous. If using shelter, keep it high, open-sided, stable, and well away from heat and flames.
If you need a raincoat for the sausages, you probably need a backup plan.
Best move
Wait for a lower-rain window, cook the awkward bits indoors if needed, and use the BBQ for finishing once the sky stops auditioning for drama.